Existence and Convergence of Solutions to Fractional Pure Critical Exponent Problems

Víctor Hernández-Santamaría; Alberto Saldaña

doi:10.1515/ans-2021-2041

Open Access Published by De Gruyter July 28, 2021

Existence and Convergence of Solutions to Fractional Pure Critical Exponent Problems

Víctor Hernández-Santamaría and Alberto Saldaña

From the journal Advanced Nonlinear Studies

https://doi.org/10.1515/ans-2021-2041

Abstract

We study existence and convergence properties of least-energy symmetric solutions (l.e.s.s.) to the pure critical exponent problem

( - Δ ) s ⁢ u s = | u s | 2 s ⋆ - 2 ⁢ u s , u s ∈ D 0 s ⁢ ( Ω ) , 2 s ⋆ := 2 ⁢ N N - 2 ⁢ s ,

where s is any positive number, Ω is either ℝN or a smooth symmetric bounded domain, and D0s⁢(Ω) is the homogeneous Sobolev space. Depending on the kind of symmetry considered, solutions can be sign-changing. We show that, up to a subsequence, a l.e.s.s. us converges to a l.e.s.s. ut as s goes to any t>0. In bounded domains, this convergence can be characterized in terms of an homogeneous fractional norm of order t-ε. A similar characterization is no longer possible in unbounded domains due to scaling invariance and an incompatibility with the functional spaces; to circumvent these difficulties, we use a suitable rescaling and characterize the convergence via cut-off functions. If t is an integer, then these results describe in a precise way the nonlocal-to-local transition. Finally, we also include a nonexistence result of nontrivial nonnegative solutions in a ball for any s>1.

Keywords: Equivariant Solutions; Symmetric Concentration Compactness; Higher-Order Fractional Laplacian; Asymptotic Analysis

MSC 2010: 35B33; 35B40; 35R11; 35J35; 35J40

1 Introduction

In this paper, we study existence and convergence properties of solutions to pure critical problems such as

(1.1) ( - Δ ) s ⁢ u s = | u s | 2 s ⋆ - 2 ⁢ u s , u s ∈ D 0 s ⁢ ( Ω ) , 2 s ⋆ := 2 ⁢ N N - 2 ⁢ s ,

where N≥1, s>0, N>2⁢s, (-Δ)s is the (possibly higher-order) fractional Laplacian, Ω is either ℝN or a smooth bounded domain of ℝN, and D0s⁢(Ω) is the homogeneous (fractional) Sobolev space, namely the closure of Cc∞⁢(Ω) with respect to the Gagliardo seminorm ∥⋅∥s, given by (2.2) below. See Section 2 for precise definitions and main properties of the operator (-Δ)s and the space D0s⁢(Ω).

Problem (1.1) is an important paradigm in nonlinear analysis of PDEs and plays an important role in the study of the well-known Yamabe problem in differential geometry and its generalizations. Moreover, the fractional Laplacian plays an important role in the study of anomalous and nonlocal diffusion, which appears for instance in continuum mechanics, graph theory, and ecology; see [16] and the references therein.

For s=1, there is an extensive literature on the existence of solutions of (1.1) using different methods; see, for instance, [35, 26, 38, 24, 25, 18, 30, 21] and the references therein. When s∈ℕ with s≥2, equation (1.1) is the pure critical exponent problem for the polyharmonic operator, and the existence of solutions has been studied in [28, 37, 9, 33, 8, 34]. In the fractional setting, existence results in ℝN for s∈(0,1) are available in [41, 23, 32, 29, 7, 46], and for s>1 it is known that (1.1) has a family of radially symmetric solutions in ℝN; see [17].

The first objective of this paper is to present a unified approach to show the existence of solutions of (1.1) for any s∈(0,∞). For Ω bounded, this is a problem that depends strongly on the geometry of the domain, whereas for Ω=ℝN all positive solutions of (1.1) are completely characterized, and therefore we are particularly interested in nonradial sign-changing entire solutions. The second objective is to investigate the convergence properties of solutions, namely if (usk)k∈ℕ are solutions of (1.1) (with sk∈(0,∞) instead of s), then what can be said about the limit of usk as sk→s0>0. For the critical nonlinearity f⁢(u)=|u|2s⋆-2⁢u, we are not aware of any previous result in this direction.

Although problem (1.1) has a variational structure (with energy functional given by (2.8)), variational methods face several compactness issues, mainly due to the following scaling invariance:

(1.2) ∥ u ∥ s = ∥ u λ , ξ ∥ s , ∫ ℝ N | u | 2 s ⋆ = ∫ ℝ N | u λ , ξ | 2 s ⋆ , u λ , ξ ⁢ ( x ) := λ N 2 - s ⁢ u ⁢ ( λ ⁢ x + ξ ) ,

for u∈D0s⁢(ℝN), λ>0, and ξ∈ℝN.

One way to overcome this difficulty is to search for solutions within a symmetric framework. In this way, we regain some compactness to achieve least-energy solutions (among symmetric functions) and we also obtain directly important information about the shape of solutions, which can be used to guarantee multiplicity results.

Following the framework from [18, 19, 20], let us introduce some notation. Let G be a closed subgroup of the group O⁢(N) of linear isometries of ℝN such that the following condition holds:

For each x∈ℝN, either dim⁡(G⁢x)>0 or G⁢x={x}, where G⁢x:={g⁢x:g∈G} is the G-orbit of x.

Let ϕ:G→ℤ2:={-1,1} be a continuous homomorphism of groups (i.e., ϕ⁢(g∘h)=ϕ⁢(g)⁢ϕ⁢(h)) and let Ω be a G-invariant set of ℝN (i.e., G⁢x⊂Ω if x∈Ω). A function u:Ω→ℝ is said to be ϕ-equivariant if

(1.3) u ⁢ ( g ⁢ x ) = ϕ ⁢ ( g ) ⁢ u ⁢ ( x ) for all ⁢ g ∈ G ⁢ and ⁢ x ∈ Ω .

Depending on ϕ, it could happen that (1.3) is only satisfied by u≡0, for instance, if G=O⁢(N) and ϕ⁢(g) is the determinant of g∈G. To avoid this, we need to impose the following condition on ϕ:

There exists ξ∈ℝN such that

{ g ∈ G : g ⁢ ξ = ξ } ⊂ ker ⁡ ϕ := { g ∈ G : ϕ ⁢ ( g ) = 1 } .

Under this condition, the space

(1.4) D 0 s ⁢ ( Ω ) ϕ := { u ∈ D 0 s ⁢ ( Ω ) : u ⁢ is ϕ -equivariant }

has infinite dimension; see [13, Theorem 3.1].

Our first result concerns bounded domains. Let ΩG:={x∈Ω:g⁢x=x⁢ for all ⁢g∈G} be the set of G-fixed points of Ω and let ℕ0:=ℕ∪{0}.

Theorem 1.1.

Assume that G and ϕ verify assumptions (A1) and (A2). Let N≥1 and let Ω⊂RN be a smooth bounded G-invariant domain such that ΩG=∅.

(Existence.) For every s > 0 with N > 2 ⁢ s , there is a ϕ -equivariant least-energy solution us of

(1.5) ( - Δ ) s ⁢ u s = | u s | 2 s ⋆ - 2 ⁢ u s , u s ∈ D 0 s ⁢ ( Ω ) ϕ ∖ { 0 } .

The solution is sign-changing if ϕ : G → { - 1 , 1 } is surjective.
(Convergence.) Let ( s k ) k ∈ ℕ ⊂ ( 0 , N 2 ) such that s k → s as k → ∞ with N > 2 ⁢ s > 0 , and let u s k be a ϕ -equivariant least-energy solution of

( - Δ ) s k ⁢ u s k = | u s k | 2 s k ⋆ - 2 ⁢ u s k , u s k ∈ D 0 s k ⁢ ( Ω ) ϕ ∖ { 0 } .

Then, up to a subsequence, there is a ϕ -equivariant least-energy solution us of (1.5) such that

(1.6) u s k → u s in ⁢ D 0 t ⁢ ( Ω ) ⁢ as ⁢ k → ∞ ⁢ for all ⁢ t ∈ [ 0 , s ) .

Different choices for G and ϕ in Theorem 1.1 produce different kinds of solutions. For instance, if G=O⁢(N) and ϕ≡1, then Theorem 1.1 yields a solution us of (1.5) which is radially symmetric. On the other hand, if G=Gi and ϕ=ϕi are given as in (2.15) and (2.16) below, then Theorem 1.1 guarantees that problem (1.5) has at least ⌊N4⌋ nonradial sign-changing solutions, where ⌊x⌋ denotes the greatest integer less than or equal to x; this existence result is new even in the local case for s∈ℕ with s≥3 (the cases s=1 and s=2 are shown in [18] and [20], respectively).

To prove the first part of Theorem 1.1 (existence), we extend to the fractional setting the strategy used in [18, 19, 20] for local problems, where a symmetric-concentration compactness argument is used. The main difficulty in this extension is the adaptation of a Brezis–Kato-type argument which is based on direct calculations for the Laplacian. Direct computations are much harder in nonlocal problems (especially in the higher-order regime s>1). We overcome this difficulty using interpolation inequalities and sharp Hardy–Littlewood–Sobolev inequalities among other tools; see Section 3.

The proof of the convergence result relies strongly on the ϕ-equivariance of the solutions, which yields the necessary compactness to extract a convergent subsequence and to guarantee that the limit is a least-energy ϕ-equivariant solution as well. We remark that (1.6) also holds in the standard Sobolev norm ∥⋅∥Hs-δ, which is equivalent to the homogeneous norm ∥⋅∥s-δ in D0s-δ⁢(Ω) with Ω bounded. After Theorem 1.3, we comment more on these results and compare our findings with previously known convergence results for subcritical problems.

The assumption ΩG=∅ is fundamental since the existence of solutions of critical problems is closely related to the geometry of the domain. Indeed, a consequence of the Pohozaev identity is that, if Ω is star-shaped and s=1, then (1.5) only admits trivial solutions. Although for any s∈(0,∞) there are versions of the Pohozaev identity (see [44, Corollary 1.7]), a general nonexistence result as in the case s=1 is, as far as we know, not available for (1.5) if s≠1. This is because the nonexistence proof also requires a unique continuation principle and the existence of a suitable extension of the solution to ℝN.

Using maximum principles, one can show nonexistence of nonnegative solutions in starshaped domains for (1.5) if s∈(0,1)∪{2}; see [43, Corollary 1.3] and [33, Theorem 7.33]. On balls, the nonexistence of nonnegative solutions is also known for s∈ℕ; see [40] (see also [33, Theorem 7.34]). Using the Pohozaev identities from [43] and a fractional higher-order Hopf Lemma from [3], we can extend this nonexistence result to any s>1.

Proposition 1.2.

Let α∈(0,1), s>1, N>2⁢s, and let B:={x∈RN:|x|<1}. The problem

(1.7) ( - Δ ) s ⁢ u = | u | 2 s ⋆ - 2 ⁢ u , u ∈ D 0 s ⁢ ( B ) ∩ C α ⁢ ( B ¯ ) ,

does not admit nontrivial nonnegative solutions.

Maximum principles (and in particular Hopf Lemmas) do not hold in general domains if s>1; for instance, positivity preserving properties fail in ellipses for

s ∈ ( 1 , 3 2 + 3 ) ,

see [6], in dumbbell domains for s∈(m,m+1) with m odd, see [3, Theorem 1.11], and in two disjoint balls for s∈(m,m+1) with m odd, see [4, Theorem 1.1] (curiously, this last set has a positive Green’s function if s∈(m,m+1) and m is even, see [3, Theorem 1.10]).

Next we present our existence and convergence results for entire solutions, namely when Ω=ℝN. This setting is more delicate for several reasons. For the existence part, there is an inherent lack of compactness due to the scaling and translation invariance (1.2). This is controlled in our proofs using the symmetric structure of D0s⁢(ℝN)ϕ. On the other hand, the characterization of the convergence of solutions faces a problem regarding the incompatibility of the functional spaces. To be more precise, by the Sobolev inequality,

D 0 s ⁢ ( ℝ N ) = D s ⁢ ( ℝ N ) := { u ∈ L 2 s ⋆ ⁢ ( ℝ N ) : ∥ u ∥ s < ∞ }

(see Theorem 2.1 below, see also [14] for a survey on homogeneous Sobolev spaces). In particular, it is not true that Dt⁢(ℝN)⊂Ds⁢(ℝN) for t>s, as it happens in bounded domains, and therefore it is not trivial to find a suitable norm to describe the convergence properties of solutions; for instance, a characterization such as (1.6) is not possible in ℝN since us might not belong to Dt⁢(ℝN) for t≠s. This is not a problem of local smoothness, but rather an incompatibility with the decay at infinity. In the following result, we show that entire solutions converge when multiplied by an arbitrary function in Cc∞⁢(ℝN).

Theorem 1.3.

Assume that N≥1 and that G and ϕ verify assumptions (A1) and (A2).

(Existence.) For every s > 0 with N > 2 ⁢ s , there is a ϕ -equivariant least-energy solution ws of

(1.8) ( - Δ ) s ⁢ w s = | w s | 2 s ⋆ - 2 ⁢ w s , w s ∈ D s ⁢ ( ℝ N ) ϕ ∖ { 0 } .

The solution is sign-changing if ϕ : G → { - 1 , 1 } is surjective.
(Convergence.) Let ( s k ) k ∈ ℕ ∈ ( 0 , N 2 ) such that s k → s as k → ∞ with N > 2 ⁢ s > 0 , and let w s k be a ϕ -equivariant least-energy solution of

( - Δ ) s k ⁢ w s k = | w s k | 2 s k ⋆ - 2 ⁢ w s k , w s k ∈ D s k ⁢ ( ℝ N ) ϕ ∖ { 0 } .

Then, up to a rescaled subsequence of w s k denoted by w ~ s k , there is a ϕ -equivariant least-energy solution ws of (1.8) such that

(1.9) η ⁢ w ~ s k → η ⁢ w s in ⁢ D t ⁢ ( ℝ N ) ⁢ as ⁢ k → ∞ ⁢ for all ⁢ η ∈ C c ∞ ⁢ ( ℝ N ) ⁢ and ⁢ t ∈ [ 0 , s ) .

In particular, w ~ s k → w s in L loc q ⁢ ( ℝ N ) as k → ∞ for all q ∈ [ 1 , 2 s ⋆ ) .

As in the bounded domain case, if G=O⁢(N) and ϕ≡1, then a solution ws of (1.8) is a radially symmetric function; see [17] for a study of this type of solutions. If G=Gi and ϕ=ϕi are those given in (2.15) and (2.16), then Theorem 1.3 yields the existence of at least ⌊N4⌋ nonradial sign-changing solutions to (1.8). For s∈(0,1), this existence result was proved in the recent paper [46], for s=1 it was shown in [18], and for s=2 it is a particular case of [20, Theorem 1.1]. All these papers follow a strategy based on a symmetric-concentration compactness argument, but at a technical level they have important differences and none of them can be easily extended to guarantee existence of solutions in the whole higher-order range s∈(1,∞). In this sense, the method we present here is more flexible and universal. We emphasize that the solutions given by Theorem 1.3 are different from those obtained in [26] for s=1, in [8] for s∈ℕ, and in [29, 7] for s∈(0,1).

These entire solutions are obtained employing a suitable rescaling of a concentrating energy-minimizing sequence, see Theorem 4.1, where it is also shown that the concentration point is necessarily a G-fixed point. See also [18, Theorem 2.5] for other variants of these types of results for the Laplacian.

In the convergence part in Theorem 1.3, the rescaling w~sk of the sequence wsk is needed to avoid the scaling invariance (1.2) typical in critical problems. Without this rescaling, it can happen that wsk converges to 0 or that it diverges at every point. A particularly useful rescaling is presented in Theorem 6.3 via condition (6.3), which is convenient for technical reasons. The use of cut-off functions to characterize the convergence (1.9) is one of the main methodological contributions of this work and it requires delicate uniform estimates.

As far as we know, Theorems 1.1 and 1.3 are the first results to consider the convergence of solutions in the critical regime (right-hand side |u|2s⋆-2⁢u) and for higher-order problems (s∈(1,∞)). Previous convergence results were only available for subcritical problems (where the compactness of the embedding D0s⁢(Ω)↪Lp⁢(Ω), 0<p<2s⋆, is the main tool) and only for sk↗1; see [11, 12, 31]. For linear problems, the continuity of the solution map s↦vs is considered in [10] as s↗1, and the continuity and differentiability of this map at any s∈(0,1) is studied in [39].

Furthermore, we mention that our convergence characterizations (1.6) and (1.9) are stronger than those of [11, 12, 31], which are in terms of L2 and Lloc2 norms. Note that solutions of nonlinear equations with a potential (such as those considered in [11, 12, 31]) would have L2 as a common space for all solutions regardless of the value of s, but this is not the case for the pure critical exponent problem (1.8).

The paper is organized as follows. In Section 2, we detail our symmetry setting and functional framework and exhibit a family of symmetry groups Gi and surjective homomorphism ϕi that, together with Theorems 1.1 and 1.3, yield nonradial sign-changing solutions. Section 3 contains the main technical tools used to show our main existence and convergence results. In Section 4, we show a concentration result using a symmetric-concentration compactness argument. Sections 5 is devoted to the proof of Theorem 1.1 and the nonexistence result stated in Proposition 1.2. Finally, in Section 6, we provide the proof of Theorem 1.1.

2 Preliminaries

In this section, we introduce the symmetric setting and functional framework to study the pure critical exponent problem (1.1). We also detail the definition and some properties of the (possibly higher-order) fractional Laplacian and the homogeneous (fractional) Sobolev space.

2.1 Functional Framework

For u∈Cc∞⁢(ℝN), the fractional Laplacian of order 2⁢σ is given by

( - Δ ) σ ⁢ u ⁢ ( x ) = c N , σ ⁢ p . v . ∫ ℝ N u ⁢ ( x ) - u ⁢ ( y ) | x - y | N + 2 ⁢ σ ⁢ d y = c N , σ ⁢ lim ε → 0 ⁡ ∫ { | x - y | > ε } u ⁢ ( x ) - u ⁢ ( y ) | x - y | N + 2 ⁢ σ ⁢ d y for ⁢ x ∈ ℝ N ,

where p.v. means the principal value sense,

(2.1) c N , σ := 4 σ ⁢ π - N 2 ⁢ σ ⁢ ( 1 - σ ) ⁢ Γ ⁢ ( N 2 + σ ) Γ ⁢ ( 2 - σ )

is a normalization constant, and Γ is the usual gamma function. Let s=m+σ>1 with m∈ℕ and σ∈(0,1). The fractional Laplacian of order 2⁢s is given by

( - Δ ) s ⁢ u ⁢ ( x ) := { ( - Δ ) m 2 ⁢ ( - Δ ) σ ⁢ ( - Δ ) m 2 ⁢ u ⁢ ( x ) for m even, ∑ i = 1 N ( - Δ ) m - 1 2 ( ∂ i ( - Δ ) σ ( ∂ i ( - Δ ) m - 1 2 u ( x ) ) ) for m odd.

We remark that other pointwise evaluations of (-Δ)s are possible, see for example [5, 45], and we refer to [1, 2, 3] for recent studies on boundary value problems involving higher-order fractional Laplacians.

For s>0, let

H s ⁢ ( ℝ N ) := { u ∈ L 2 ⁢ ( ℝ N ) : ( 1 + | ξ | 2 ) s 2 ⁢ u ^ ∈ L 2 ⁢ ( ℝ N ) }

denote the usual fractional Sobolev space, where u^ denotes the Fourier transform of u. For Ω⊂ℝN a smooth open set, let D0s⁢(Ω) be the closure of Cc∞⁢(Ω) with respect to the norm

(2.2) ∥ u ∥ s := ( ℰ s ⁢ ( u , u ) ) 1 2 ,

where

(2.3) ℰ s ⁢ ( u , v ) = ∫ ℝ N | ξ | 2 ⁢ s ⁢ u ^ ⁢ ( ξ ) ⁢ v ^ ⁢ ( ξ ) ⁢ d ξ

is the associated scalar product. If Ω=ℝN, then we simply write Ds⁢(ℝN) instead of D0s⁢(ℝN). Let

ℋ 0 s ⁢ ( Ω ) := { u ∈ H s ⁢ ( ℝ N ) : u = 0 ⁢ on ⁢ ℝ N ∖ Ω }

be equipped with the standard Hs-norm. If Ω is bounded, then

(2.4) ℓ 1 , s ⁢ ∥ u ∥ ℋ 0 s ⁢ ( Ω ) ≤ ∥ u ∥ s ≤ ∥ u ∥ ℋ 0 s ⁢ ( Ω ) ,

where ℓ1,s=2-1⁢min⁡{1,λs,1} and λ1,s=λ1,s⁢(Ω) is the first eigenvalue of ((-Δ)s,ℋ0s⁢(Ω)); see, for example, [4].

If m∈ℕ, σ∈(0,1), s=m+σ, and u,v∈Ds⁢(ℝN), then the following are equivalent expressions for ℰs:

ℰ σ ⁢ ( u , v ) = c N , σ 2 ⁢ ∫ ℝ N ∫ ℝ N ( u ⁢ ( x ) - u ⁢ ( y ) ) ⁢ ( v ⁢ ( x ) - v ⁢ ( y ) ) | x - y | N + 2 ⁢ σ ⁢ d x ⁢ d y ,

(2.5) ℰ s ⁢ ( u , v ) = { ℰ σ ⁢ ( ( - Δ ) m 2 ⁢ u , ( - Δ ) m 2 ⁢ v ) if m is even, ∑ k = 1 N ℰ σ ( ∂ k ( - Δ ) m - 1 2 u , ∂ k ( - Δ ) m - 1 2 v ) if m is odd.

For some results we also consider s∈ℕ, in these cases we have that

ℰ s ⁢ ( u , v ) = { ∫ ℝ N ( - Δ ) m 2 ⁢ u ⁢ ( - Δ ) m 2 ⁢ v if m is even, ∫ ℝ N ∇ ( - Δ ) m - 1 2 u ∇ ( - Δ ) m - 1 2 v if m is odd.

In any case, for s>0,

∫ ℝ N ( - Δ ) s ⁢ u ⁢ ( x ) ⁢ v ⁢ ( x ) ⁢ d x = ℰ s ⁢ ( u , v ) for ⁢ u ∈ C c ∞ ⁢ ( ℝ N ) ⁢ and ⁢ v ∈ D s ⁢ ( ℝ N ) ;

see, for example, [5, 4]. Throughout this paper, the Lq-norm is denoted by

| f | q := ( ∫ Ω | f ⁢ ( x ) | q ⁢ d x ) 1 q for ⁢ q ∈ [ 1 , ∞ ) .

We close this section with two important results.

Theorem 2.1 (Fractional Sobolev Inequality).

Let N≥1, s>0, and N>2⁢s. Then there is κN,s>0 such that |u|2s⋆≤κN,s⁢∥u∥s for all u∈Ds⁢(RN), where

(2.6) κ N , s = 2 - 2 ⁢ s ⁢ π - s ⁢ Γ ⁢ ( N - 2 ⁢ s 2 ) Γ ⁢ ( N + 2 ⁢ s 2 ) ⁢ ( Γ ⁢ ( N ) Γ ⁢ ( N 2 ) ) 2 ⁢ s N .

Proof.

See [22, Theorem 1.1]. ∎

Theorem 2.2.

Let Ω be a bounded smooth domain and let s>0, N>2⁢s, p∈[1,2s⋆), and ε∈(0,s]. Then the embeddings D0s⁢(Ω)↪D0s-ε⁢(Ω) and D0s⁢(Ω)↪Lp⁢(Ω) are compact.

Proof.

The compactness of the embedding ℋ0s⁢(Ω)↪ℋ0s-ε⁢(Ω) follows by interpolation theory; see [47]. The space ℋ0t⁢(Ω) is defined in [47, Section 4.3.2, (1a)]. That A=ℋ0s⁢(Ω) is an interpolation space between A0=H0⌈s⌉⁢(Ω) and A1=L2⁢(Ω) is a consequence of [47, Section 4.3.2, Theorem 2] together with [47, Section 2.4.2, (10)]. Finally, the compactness of the embedding ℋ0s⁢(Ω)↪ℋ0s-ε⁢(Ω) follows from [47, Section 1.16.4, Theorem 2 (a)] together with the compactness of the embedding A0↪A1; see [36, Section 7.10]. Then the compactness of D0s⁢(Ω)↪D0s-ε⁢(Ω) holds by the equivalence of norms (2.4). The embedding D0s⁢(Ω)↪Lp⁢(Ω) is compact for p∈[1,2s⋆) by [22, Theorem 1.5]. ∎

2.2 Symmetric Setting

Following [18, 19, 20] we now present a series of results connecting the symmetric framework presented in the introduction and the variational structure of equation (1.1).

Let G be a closed subgroup of O⁢(N) and let ϕ:G→ℤ2:={-1,1} be a continuous homomorphism of groups satisfying the properties (A1) and (A2) presented in the introduction. Let Ω be a G-invariant bounded smooth domain of ℝN and recall the definition of ϕ-equivariance given in (1.3) and of the space D0s⁢(Ω)ϕ given in (1.4). We say that u∈D0s⁢(Ω) is a solution of

(2.7) ( - Δ ) s ⁢ u = | u | 2 s ⋆ - 2 ⁢ u , u ∈ D 0 s ⁢ ( Ω ) ,

if u is a critical point of the C1-functional Js:D0s⁢(Ω)→ℝ defined by

(2.8) J s ⁢ ( u ) := 1 2 ⁢ ∥ u ∥ s 2 - 1 2 s ⋆ ⁢ | u | 2 s ⋆ 2 s ⋆ .

The next lemma is a type of principle of symmetric criticality in the ϕ-equivariant setting, and it extends [20, Lemma 3.1] to the fractional setting. For φ∈Cc∞⁢(Ω), let

(2.9) φ ϕ ⁢ ( x ) := 1 μ ⁢ ( G ) ⁢ ∫ G ϕ ⁢ ( g ) ⁢ φ ⁢ ( g ⁢ x ) ⁢ d μ ,

where μ is the Haar measure on G. In particular, φϕ∈Cc∞⁢(Ω)ϕ.

Lemma 2.3.

Let m∈N0 and σ∈[0,1] be such that s:=m+σ>0. If u∈D0s⁢(Ω)ϕ, then

J s ′ ⁢ ( u ) ⁢ φ ϕ = J s ′ ⁢ ( u ) ⁢ φ for every ⁢ φ ∈ C c ∞ ⁢ ( Ω ) .

Moreover, if Js′⁢(u)⁢ϑ=0 for every ϑ∈Cc∞⁢(Ω)ϕ, then Js′⁢(u)⁢φ=0 for every φ∈Cc∞⁢(Ω).

Proof.

We show the case m∈ℕ0 even and σ∈(0,1). The other cases follow analogously. First, notice that

( - Δ ) m 2 ⁢ ( v ∘ g ) = ( - Δ ) m 2 ⁢ v ∘ g for every ⁢ v ∈ D 0 s ⁢ ( Ω ) ⁢ and ⁢ g ∈ G .

So, if u is ϕ-equivariant, then we have that (-Δ)m2⁢u is ϕ-equivariant too. Also,

(2.10) ( - Δ ) m 2 ⁢ ( φ ϕ ) ⁢ ( x ) = 1 μ ⁢ ( G ) ⁢ ∫ G ( - Δ ) m 2 ⁢ ( ϕ ⁢ ( g ) ⁢ φ ∘ g ) ⁢ ( x ) ⁢ d μ = 1 μ ⁢ ( G ) ⁢ ∫ G ϕ ⁢ ( g ) ⁢ ( - Δ ) m 2 ⁢ φ ⁢ ( g ⁢ x ) ⁢ d μ

and

(2.11) J s ′ ( u ) φ ϕ = ℰ σ ( ( - Δ ) m 2 u , ( - Δ ) m 2 φ ϕ ) - ∫ Ω | u ( x ) | p - 2 u ( x ) φ ϕ ( x ) d x = : J 1 + J 2 .

For J2, we use that u is ϕ-equivariant to obtain that

(2.12) J 2 = 1 μ ⁢ ( G ) ⁢ ∫ Ω ∫ G | u ⁢ ( x ) | p - 2 ⁢ u ⁢ ( x ) ⁢ ϕ ⁢ ( g ) ⁢ φ ⁢ ( g ⁢ x ) ⁢ d μ ⁢ d x = ∫ Ω | u ⁢ ( y ) | p - 2 ⁢ u ⁢ ( y ) ⁢ φ ⁢ ( y ) ⁢ d y .

For J1, we argue as follows. Since φϕ∈Cc∞⁢(Ω)ϕ, we use (2.10) to obtain that

∫ ℝ N ∫ ℝ N [ ( - Δ ) m 2 ⁢ u ⁢ ( x ) - ( - Δ ) m 2 ⁢ u ⁢ ( y ) ] ⁢ [ ∫ G ϕ ⁢ ( g ) ⁢ ( ( - Δ ) m 2 ⁢ φ ⁢ ( g ⁢ x ) - ( - Δ ) m 2 ⁢ φ ⁢ ( g ⁢ y ) ) ⁢ 𝑑 μ ] | x - y | N + 2 ⁢ σ ⁢ d x ⁢ d y

= ∫ ℝ N ∫ ℝ N ∫ G [ ( - Δ ) m 2 ⁢ u ⁢ ( g ⁢ x ) - ( - Δ ) m 2 ⁢ u ⁢ ( g ⁢ y ) ] ⁢ [ ( - Δ ) m 2 ⁢ φ ⁢ ( g ⁢ x ) - ( - Δ ) m 2 ⁢ φ ⁢ ( g ⁢ y ) ] | x - y | N + 2 ⁢ σ ⁢ d μ ⁢ d x ⁢ d y .

By setting the change of variable x¯=g⁢x (resp. y¯=g⁢y) and using Fubini’s theorem, we have that, for every g∈G,

J 1 = c N , σ μ ⁢ ( G ) ⁢ ∫ G ∫ ℝ N ∫ ℝ N [ ( - Δ ) m 2 ⁢ u ⁢ ( x ¯ ) - ( - Δ ) m 2 ⁢ u ⁢ ( y ¯ ) ] ⁢ [ ( - Δ ) m 2 ⁢ φ ⁢ ( x ¯ ) - ( - Δ ) m 2 ⁢ φ ⁢ ( y ¯ ) ] | g - 1 ⁢ ( x ¯ - y ¯ ) | N + 2 ⁢ σ ⁢ d x ⁢ d y ⁢ d μ

(2.13) = c N , σ ⁢ ∫ ℝ N ∫ ℝ N [ ( - Δ ) m 2 ⁢ u ⁢ ( x ¯ ) - ( - Δ ) m 2 ⁢ u ⁢ ( y ¯ ) ] ⁢ [ ( - Δ ) m 2 ⁢ φ ⁢ ( x ¯ ) - ( - Δ ) m 2 ⁢ φ ⁢ ( y ¯ ) ] | x ¯ - y ¯ | N + 2 ⁢ σ ⁢ d x ¯ ⁢ d y ¯ .

To conclude, it is enough to collect identities (2.12) and (2.13) and insert into (2.11) to deduce that

J s ′ ⁢ ( u ) ⁢ φ ϕ = J s ′ ⁢ ( u ) ⁢ φ .

The rest of the proof follows immediately. ∎

As a consequence of the previous results, the nontrivial ϕ-equivariant solutions of problem (2.7) belong to the Nehari set

(2.14) 𝒩 s ϕ ⁢ ( Ω ) := { u ∈ D 0 s ⁢ ( Ω ) ϕ : u ≠ 0 , ∥ u ∥ s 2 = | u | 2 s ⋆ 2 s ⋆ } .

Let

c s ϕ ⁢ ( Ω ) := inf u ∈ 𝒩 s ϕ ⁢ ( Ω ) ⁡ J s ⁢ ( u ) .

The following result gives some properties of 𝒩sϕ⁢(Ω) and csϕ⁢(Ω).

Lemma 2.4.

Let s>0.

There exists a 0 > 0 such that ∥ u ∥ s ≥ a 0 for every u ∈ 𝒩 s ϕ ⁢ ( Ω ) .
𝒩 s ϕ ⁢ ( Ω ) is a C 1 -Banach sub-manifold of D 0 s ⁢ ( Ω ) and a natural constraint for J s .
Let

𝒯 := { σ ∈ C 0 ⁢ ( [ 0 , 1 ] ; D 0 s ⁢ ( Ω ) ϕ ) : σ ⁢ ( 0 ) = 0 , σ ⁢ ( 1 ) ≠ 0 , J s ⁢ ( σ ⁢ ( 1 ) ) ≤ 0 } .

Then

c s ϕ ⁢ ( Ω ) = inf σ ∈ 𝒯 ⁡ max t ∈ [ 0 , 1 ] ⁡ J s ⁢ ( σ ⁢ ( t ) ) .

Proof.

The proof follows exactly as in [19, Lemma 2.1] using Theorem 2.1. ∎

For a G-invariant domain Ω, let us denote by ΩG the set of G-fixed points in Ω, more precisely,

Ω G := { x ∈ Ω : G ⁢ x = { x } } .

The next result characterizes the least-energy level on domains with G-fixed points.

Lemma 2.5.

Let s>0. If ΩG≠∅, then csϕ⁢(Ω)=csϕ⁢(RN).

Proof.

From the inclusion D0s⁢(Ω)⊂Ds⁢(ℝN), we have that 𝒩sϕ⁢(Ω)⊂𝒩sϕ⁢(ℝN). Then

c s ϕ ⁢ ( ℝ N ) = inf 𝒩 s ϕ ⁢ ( ℝ N ) ⁡ J s ≤ inf 𝒩 s ϕ ⁢ ( Ω ) ⁡ J s = c s ϕ ⁢ ( Ω ) .

For the converse, consider a sequence (φk)k in 𝒩sϕ⁢(ℝN)∩Cc∞⁢(ℝN) such that Js⁢(φk)→csϕ⁢(ℝN) and let x0∈ΩG and λk>0 such that

φ k ⋆ ⁢ ( x ) := λ k - N 2 + s ⁢ φ k ⁢ ( λ k - 1 ⁢ ( x - x 0 ) )

has support in Ω. As x0 is a G-fixed point, φk⋆ is ϕ-equivariant. Thus φk⋆∈𝒩ϕ⁢(Ω), and hence

c s ϕ ⁢ ( Ω ) ≤ J s ⁢ ( φ k ⋆ ) = J s ⁢ ( φ k ) for all ⁢ k .

Letting k→+∞, we conclude that csϕ⁢(Ω)≤csϕ⁢(ℝN). This ends the proof. ∎

The following lemma can be found in [19, Lemma 2.4] or [20, Lemma 3.4].

Lemma 2.6.

If G satisfies (A1), then, for every pair of sequences (λk)k∈N⊂(0,∞) and (xk)k∈N⊂RN, there exist C0>0 and (ξk)k∈N⊂RN such that, up to a subsequence, λk-1⁢dist⁡(G⁢xk,ξk)≤C0 for all k∈N. Moreover, one of the following statements hold true: either ξk∈ΩG or, for each m∈N, there exist g1,…,gm∈G such that λk-1⁢|gi⁢ξk-gj⁢ξk|→∞ as k→∞ if i≠j.

2.3 Groups and Homomorphism for Sign-Changing Solutions

In this section, we present some symmetry groups and surjective homomorphisms that can be used to obtain different sign-changing ϕ-equivariant solutions. These groups and homomorphism were also used in [19, Lemma 3.2] and [20, Lemma 4.2].

Let rθ:ℝ2→ℝ2 be a rotation matrix (counterclockwise through an angle θ∈[0,π)) and for

x = ( x 1 , x 2 , x 3 , x 4 ) T ∈ ℝ 4

let Rθ,ρ:ℝ4→ℝ4 be given by

R θ ⁢ x := ( r θ 0 0 r θ ) ⁢ x and ρ ⁢ x := ( x 3 , x 4 , x 1 , x 2 ) T .

Let Υ be the subgroup generated by {Rθ,ρ:θ∈[0,2⁢π)} and let ϕ:Υ→ℤ2 be the homomorphism given by ϕ⁢(Rθ):=1 for any θ∈[0,2⁢π) and ϕ⁢(ρ)=-1. For N≥4, let n:=⌊N4⌋≥1, Λj:=O⁢(N-4) if j=1,…,n-1, and Λn:={1}. The Λj-orbit of a point y∈ℝN-4⁢j is an (N-4⁢j-1)-dimensional sphere if j=1,…,n-1, and it is a single point if j=n. Define

(2.15) G j := ( Υ ) j × Λ j

acting on ℝN=ℝ4⁢j×ℝN-4⁢j by

( γ 1 , … , γ j , η ) ⁢ ( x 1 , … , x j , y ) := ( γ 1 ⁢ x 1 , … , γ j ⁢ x j , η ⁢ y ) ,

where γi∈Υ, η∈Λj, and xi∈ℝ4, and let

(2.16) ϕ j : G j → ℤ 2 be the homomorphism ϕ j ⁢ ( γ 1 , … , γ j , η ) := ϕ ⁢ ( γ 1 ) ⁢ ⋯ ⁢ ϕ ⁢ ( γ j ) .

The Gj-orbit of (z1,…,zj,y) is the product of orbits Gj⁢(z1,…,zj,y)=Υ⁢z1×⋯×Υ⁢zj×Λj⁢y.

Note that ϕj is surjective and (A1) and (A2) are satisfied by Gj and ϕj for each j=1,…,n. Moreover, if u is ϕi-equivariant, v is ϕj-equivariant with i<j, and u⁢(x)=v⁢(x)≠0 for some x=(z1,…,zj,y)∈ℝN, then, as u⁢(z1,…,ρ⁢zj,y)=u⁢(z1,…,zj,y) and v⁢(z1,…,ρ⁢zj,y)=-v⁢(z1,…,zj,y), we have that

u ⁢ ( z 1 , … , ρ ⁢ z j , y ) ≠ v ⁢ ( z 1 , … , ρ ⁢ z j , y ) .

As a consequence, u≠v, and Theorems 1.1 and 1.3 yield the existence of at least ⌊N4⌋ nonradial sign-changing solutions, where ⌊x⌋ denotes the greatest integer less than or equal to x.

We refer to [20, Remark 4.3] for an example of a ϕj-equivariant function and for an explanation on why a similar construction is impossible for N=1,2,3.

3 Uniform Bounds and Asymptotic Estimates

Let Ω⊂ℝN be a smooth bounded domain.

Lemma 3.1.

Let s>0, δ∈(0,s), sk∈(s-δ2,s+δ2), and let uk∈D0sk⁢(Ω) for k∈N. There is C>0 depending only on s, Ω, and δ such that ∥uk∥s-δ≤C⁢∥uk∥sk for all k∈N.

Proof.

We argue as in [39, Lemma 5.1]. By (2.3) and (2.2),

∥ u k ∥ s - δ 2 = ℰ s - δ ⁢ ( u k , u k ) = ∫ ℝ N | ξ | 2 ⁢ ( s - δ ) ⁢ | u ^ k | 2 ⁢ d ξ ≤ ε 2 ⁢ ( s - δ ) ⁢ ∥ u k ∥ L 2 ⁢ ( ℝ N ) 2 + ∫ | ξ | ≥ ε | ξ | 2 ⁢ ( s - δ ) ⁢ | u ^ k | 2 ⁢ d ξ

for any ε∈(0,1], where u^k is the Fourier transform of uk. Then, using that sk<s+δ2, we obtain

(3.1) ∥ u k ∥ s - δ 2 ≤ ε 2 ⁢ ( s - δ ) ⁢ ∥ u k ∥ L 2 ⁢ ( ℝ N ) 2 + ε 2 ⁢ ( s - δ - s k ) ⁢ ∫ ℝ N | ξ | 2 ⁢ s k ⁢ | u ^ k | 2 ⁢ d ξ ≤ ε 2 ⁢ ( s - δ ) ⁢ ∥ u k ∥ L 2 ⁢ ( ℝ N ) 2 + ε - δ ⁢ ∥ u k ∥ s k 2 .

Since sk>s-δ2, we have, by Theorem 2.2, that uk∈D0s-δ⁢(Ω) and, by the fractional Poincaré inequality (see, e.g., [4, Proposition 3.3]) and (2.4), there exists C0>0 depending only on s, δ, and Ω such that |uk|22≤C0⁢∥uk∥s-δ2. Fix

ε = min ⁡ { 1 , ( 1 2 ⁢ C 0 ) 1 2 ⁢ ( s - δ ) } .

Then, by (3.1), ∥uk∥s-δ2≤C⁢∥uk∥sk2, where C=2⁢ε-δ depends only on s, Ω, and δ. ∎

Lemma 3.2.

Let (sk)k∈N⊂(0,∞) be such that sk→s=m+σ as k→∞ with m∈N0 and σ∈(0,1]. Let wk∈Dsk⁢(RN) be such that

(3.2) ∥ w k ∥ s k < C for all ⁢ k ∈ ℕ ⁢ and for some ⁢ C > 0 .

Then, up to a subsequence, there is w∈Ds⁢(RN) such that

(3.3) η ⁢ w k → η ⁢ w in ⁢ D s - δ ⁢ ( ℝ N ) ⁢ as ⁢ k → ∞ ⁢ for all ⁢ η ∈ C c ∞ ⁢ ( ℝ N ) ⁢ and all ⁢ δ ∈ ( 0 , s ] .

In particular, for p∈[1,2s⋆),

(3.4) w k → w ⁢ in ⁢ L loc p ⁢ ( ℝ N ) , w k → w ⁢ a.e. in ⁢ ℝ N , w k → w ⁢ in ⁢ H loc m ⁢ ( ℝ N ) .

Proof.

Let C, δ and wk be as in the statement, and let η∈Cc∞⁢(ℝN) and K:=supp⁡η. In the following, M>0 denotes possibly different constants depending at most on C, N, s, δ, and η. Then, by Lemmas 3.1 and A.3, up to a subsequence,

∥ η ⁢ w k ∥ s - δ 2 ≤ M ⁢ ∥ η ⁢ w k ∥ s k ≤ M for all ⁢ k ∈ ℕ .

By Theorem 2.2, up to a subsequence, η⁢wk→η⁢w in Ds-δ⁢(ℝN) as k→∞, and (3.3) follows. Moreover, by (2.3) and Fatou’s Lemma,

(3.5) ∥ w ∥ s 2 = ∫ ℝ N | ξ | 2 ⁢ s ⁢ | w ^ ⁢ ( ξ ) | 2 ⁢ d ξ ≤ lim inf k → ∞ ⁡ ∫ ℝ N | ξ | 2 ⁢ s k ⁢ | w ^ k ⁢ ( ξ ) | 2 ⁢ d ξ = lim inf k → ∞ ⁡ ∥ w k ∥ s k 2 < C ,

and therefore w∈Ds⁢(ℝN). The convergence (3.4) follows from Theorem 2.2. ∎

Lemma 3.3.

For every k∈N, suppose σk∈(0,1) and wk∈Dσk⁢(RN) are such that limk→∞σk=:σ∈[0,1], ∥wk∥σk<C for all k∈N and for some C>0, and

(3.6) w k → 0 in ⁢ L loc 2 ⁢ ( ℝ N ) ⁢ as ⁢ k → ∞ .

Then, up to a subsequence,

∥ w k ⁢ η ∥ σ k 2 ≤ ℰ σ k ⁢ ( w k , η 2 ⁢ w k ) + o ⁢ ( 1 ) as ⁢ k → ∞ ⁢ for all ⁢ η ∈ C c ∞ ⁢ ( ℝ N ) .

Proof.

Let η∈Cc∞⁢(ℝN). Then

(3.7) ∥ η ⁢ w k ∥ σ k 2 - ℰ σ k ⁢ ( w k , η 2 ⁢ w k ) = c N , σ k 2 ⁢ ∫ ℝ N ∫ ℝ N w k ⁢ ( x ) ⁢ w k ⁢ ( y ) ⁢ | η ⁢ ( x ) - η ⁢ ( y ) | 2 | x - y | N + 2 ⁢ σ k ⁢ d x ⁢ d y .

Let K be the support of η and let U:={x∈ℝN:dist⁡(x,K)≤1}. In the following, C>0 denotes possibly different constants depending at most on N and η. By Fubini’s theorem, the Cauchy–Schwarz inequality, and (3.6),

≤ c N , σ k ⁢ ∫ K | w k ⁢ ( y ) | ⁢ ∫ ℝ N ∖ U | w k ⁢ ( x ) | ⁢ | η ⁢ ( y ) | 2 | x - y | N + 2 ⁢ σ k ⁢ d x ⁢ d y

≤ c N , σ k ⁢ ( ∫ K | w k | 2 ) 1 2 ⁢ ( ∫ K ( ∫ ℝ N ∖ U | w k ⁢ ( x ) | ⁢ | η ⁢ ( y ) | 2 | x - y | N + 2 ⁢ σ k ⁢ d x ) 2 ⁢ d y ) 1 2

(3.8) = o ⁢ ( 1 ) as ⁢ k → ∞ ,

where we used that cN,σk is uniformly bounded by (2.1) and we used that, by Hölder’s inequality, Theorem 2.1, and the bound ∥wk∥σk<C,

∫ K ( ∫ ℝ N ∖ U | w k ⁢ ( x ) | ⁢ | η ⁢ ( y ) | 2 | x - y | N + 2 ⁢ σ k ⁢ d x ) 2 ⁢ d y ≤ C ⁢ | w k | 2 σ k ⋆ 2 ⁢ ∫ K ( ∫ ℝ N ∖ U | x - y | - 2 ⁢ N ⁢ d x ) N + 2 ⁢ σ k N ⁢ d y < C .

On the other hand, by Fubini’s theorem,

∫ U w k ⁢ ( x ) ⁢ ∫ ℝ N w k ⁢ ( y ) ⁢ | η ⁢ ( x ) - η ⁢ ( y ) | 2 | x - y | N + 2 ⁢ σ k ⁢ d y ⁢ d x

= ∫ ℝ N w k ⁢ ( y ) ⁢ ∫ U w k ⁢ ( x ) ⁢ | η ⁢ ( x ) - η ⁢ ( y ) | 2 | x - y | N + 2 ⁢ σ k ⁢ d x ⁢ d y

(3.9) = ∫ ℝ N ∖ U w k ⁢ ( y ) ⁢ ∫ K w k ⁢ ( x ) ⁢ | η ⁢ ( x ) | 2 | x - y | N + 2 ⁢ σ k ⁢ d x ⁢ d y + ∫ U w k ⁢ ( y ) ⁢ ∫ U w k ⁢ ( x ) ⁢ | η ⁢ ( x ) - η ⁢ ( y ) | 2 | x - y | N + 2 ⁢ σ k ⁢ d x ⁢ d y .

The first summand is o⁢(1) as in (3.8). For the second summand, by the mean value theorem,

I := c N , σ k ⁢ ∫ U ∫ U w k ⁢ ( y ) ⁢ w k ⁢ ( x ) ⁢ | η ⁢ ( x ) - η ⁢ ( y ) | 2 | x - y | N + 2 ⁢ σ k ⁢ d x ⁢ d y ≤ C ⁢ c N , σ k ⁢ ∫ U × U w k ⁢ ( y ) ⁢ w k ⁢ ( x ) | x - y | N - 2 ⁢ ( 1 - σ k ) ⁢ d ⁢ ( x , y ) .

Using the Hardy–Littlewood–Sobolev inequality (see [42, Theorem 4.3] with ε=2⁢(1-σk), λ=N-ε, and p=r=2(ε/N)+1<2), equation (2.1), and the boundedness of U, we obtain

I ≤ C ⁢ c N , σ k ⁢ ( π N - 2 ⁢ ( 1 - σ k ) 2 ⁢ Γ ⁢ ( N 2 - N - 2 ⁢ ( 1 - σ k ) 2 ) Γ ⁢ ( N - N - 2 ⁢ ( 1 - σ k ) 2 ) ⁢ ( Γ ⁢ ( N 2 ) Γ ⁢ ( N ) ) N - 2 ⁢ ( 1 - σ k ) N - 1 ) ⁢ ( ∫ U | w k | p ) 1 p

≤ C ⁢ σ k ⁢ ( 1 - σ k ) ⁢ Γ ⁢ ( 1 - σ k ) ⁢ ( ∫ U | w k | p ) 1 p

(3.10) ≤ C ⁢ ( ∫ U | w k | 2 ) 1 2 = o ⁢ ( 1 ) as ⁢ k → ∞ .

The claim now follows from (3.7)–(3.10). ∎

To estimate all lower-order terms, we use the following lemma.

Lemma 3.4.

Let (σk)k∈N⊂(0,1), m∈N, and σ∈[0,1] be such that sk:=m+σk→s:=m+σ>0 as k→∞. For k∈N, let wk∈Dsk⁢(RN) be such that (3.2) holds and

(3.11) w k → 0 pointwisely in ⁢ ℝ N ⁢ as ⁢ k → ∞ .

Let α∈N0N be a multi-index such that |α|<m. Then, up to a subsequence,

∥ ψ ⁢ ∂ α ⁡ w k ∥ σ k = o ⁢ ( 1 ) as ⁢ k → ∞ ⁢ for all ⁢ ψ ∈ C c ∞ ⁢ ( ℝ N ) .

Proof.

Let ψ∈Cc∞⁢(ℝN), K=supp⁡(ψ), and let C>0 denote possibly different constants depending at most on N, m, σ, and ψ. By (3.11) and Lemma 3.2,

(3.12) ∥ w k ∥ H m ⁢ ( K ) → 0 as ⁢ k → ∞ ,

where ∥⋅∥Hm⁢(K) denotes the usual norm in the Sobolev space Hm⁢(K).

The claim now follows from the interpolation inequality (see, for example, [15, Theorem 1] using s2=1, s1=0, p1=p2=p=2, s=σk, and θ=1-σk) because

∥ ψ ⁢ ∂ α ⁡ w k ∥ σ k ≤ | ψ ⁢ ∂ α ⁡ w k | 2 1 - σ k ⁢ ∥ ψ ⁢ ∂ α ⁡ w k ∥ H 1 ⁢ ( K ) σ k = o ⁢ ( 1 ) as ⁢ k → ∞ ,

since, by (3.12),

| ψ ⁢ ∂ α ⁡ w k | 2 2 ≤ C ⁢ ∫ K | ∂ α ⁡ w k | 2 = o ⁢ ( 1 ) as ⁢ k → ∞ ,

and

∥ ψ ⁢ ∂ α ⁡ w k ∥ H 1 ⁢ ( K ) 2 = ∫ K | ∇ ⁡ ( ∂ α ⁡ w k ⁢ ψ ) | 2 ≤ C ⁢ ∫ K | ∇ ⁡ ∂ α ⁡ w k | 2 + | ∂ α ⁡ w k | 2 ≤ C ⁢ ∥ w k ∥ H m ⁢ ( K ) + o ⁢ ( 1 ) = o ⁢ ( 1 ) as ⁢ k → ∞ . ∎

Lemma 3.5.

For k∈N, let (sk)k∈N⊂(0,∞) be bounded and let wk∈Dsk⁢(RN) be such that (3.2) and (3.11) hold. Then, for any ε∈(0,1), up to a subsequence,

∥ w k ⁢ φ ∥ s k 2 ≤ ( 1 + ε ) ⁢ ℰ s k ⁢ ( w k , φ 2 ⁢ w k ) + o ⁢ ( 1 ) as ⁢ k → ∞ ⁢ for all ⁢ φ ∈ C c ∞ ⁢ ( ℝ N ) .

Proof.

Since (sk)k∈ℕ is bounded, passing to a subsequence, there is m∈ℕ0 and (σk)k∈ℕ⊂[0,1] such that limk→∞σk=:σ∈[0,1] and sk=m+σk→s=m+σ≥0 as k→∞.

Assume first that sk∈(m,m+1) and m is even. Observe that

Δ m 2 ⁢ ( w k ⁢ φ ) = φ ⁢ Δ m 2 ⁢ w k + R k ,

where Rk is a sum of products with derivatives of wk of order smaller than m. Then, by Cauchy’s inequality, for ε>0 arbitrarily small there is C⁢(ε)>0 such that

| Δ m 2 ⁢ ( w k ⁢ φ ) ⁢ ( x ) - Δ m 2 ⁢ ( w k ⁢ φ ) ⁢ ( y ) | 2 = | φ ⁢ ( x ) ⁢ Δ m 2 ⁢ w k ⁢ ( x ) - φ ⁢ ( y ) ⁢ Δ m 2 ⁢ w k ⁢ ( y ) + R ⁢ ( x ) - R ⁢ ( y ) | 2

≤ ( 1 + ε ) ⁢ | φ ⁢ ( x ) ⁢ Δ m 2 ⁢ w k ⁢ ( x ) - φ ⁢ ( y ) ⁢ Δ m 2 ⁢ w k ⁢ ( y ) | 2 + C ⁢ ( ε ) ⁢ | R ⁢ ( x ) - R ⁢ ( y ) | 2 .

Therefore, by Lemma 3.4,

(3.13) ∥ w k ⁢ φ ∥ s k 2 = ∥ Δ m 2 ⁢ ( w k ⁢ φ ) ∥ σ k 2 ≤ ( 1 + ε ) ⁢ ∥ φ ⁢ Δ m 2 ⁢ w k ∥ σ k 2 + C ⁢ ( ε ) ⁢ ∥ R ∥ σ k 2 = ( 1 + ε ) ⁢ ∥ φ ⁢ Δ m 2 ⁢ w k ∥ σ k 2 + o ⁢ ( 1 )

as k→∞. Moreover, by Lemma 3.3,

(3.14) ∥ φ ⁢ Δ m 2 ⁢ w k ∥ σ k ≤ ℰ σ k ⁢ ( Δ m 2 ⁢ w k , φ 2 ⁢ Δ m 2 ⁢ w k ) + o ⁢ ( 1 ) as ⁢ k → ∞ .

Observe that

( - Δ ) m 2 ⁢ ( φ 2 ⁢ w k ) = φ 2 ⁢ ( - Δ ) m 2 ⁢ w k + R ~ ,

where R~ has derivatives of wk with order lower than m. Then

ℰ σ k ⁢ ( Δ m 2 ⁢ w k , φ 2 ⁢ Δ m 2 ⁢ w k ) = ℰ σ k ⁢ ( Δ m 2 ⁢ w k , Δ m 2 ⁢ ( φ 2 ⁢ w k ) ) - ℰ σ k ⁢ ( Δ m 2 ⁢ w k , R ~ ) .

By (3.2), the Cauchy–Schwarz inequality, and Lemma 3.4, we have that

| ℰ σ k ⁢ ( Δ m 2 ⁢ w k , R ~ ) | ≤ ∥ w k ∥ s k ⁢ ∥ R ~ ∥ σ k = o ⁢ ( 1 ) as ⁢ k → ∞ ,

and therefore

(3.15) ℰ σ k ⁢ ( Δ m 2 ⁢ w k , φ 2 ⁢ Δ m 2 ⁢ w k ) = ℰ σ k ⁢ ( Δ m 2 ⁢ w k , Δ m 2 ⁢ ( φ 2 ⁢ w k ) ) + o ⁢ ( 1 ) as ⁢ k → ∞ .

But then, by (3.13)–(3.15),

∥ w k ⁢ φ ∥ s k 2 ≤ ( 1 + ε ) ⁢ ℰ σ k ⁢ ( Δ m 2 ⁢ w k , Δ m 2 ⁢ ( φ 2 ⁢ w k ) ) + o ⁢ ( 1 ) as ⁢ k → ∞ ,

as claimed.

The case sk∈(m,m+1) with m odd is analogous by using the corresponding norms and scalar products; see (2.5). On the other hand, if sk=m for all k∈ℕ with m even, then, by Lemma 3.2,

∥ w k ⁢ φ ∥ s k 2 = ∫ ℝ N | ( - Δ ) m 2 ⁢ ( w k ⁢ φ ) | 2

= ∫ ℝ N φ 2 ⁢ | ( - Δ ) m 2 ⁢ w k | 2 + o ⁢ ( 1 )

= ∫ ℝ N ( - Δ ) m 2 ⁢ w k ⁢ ( - Δ ) m 2 ⁢ ( φ 2 ⁢ w k ) + o ⁢ ( 1 ) as ⁢ k → ∞ .

The case sk=m for all k∈ℕ with m odd is analogous. This ends the proof. ∎

Lemma 3.6.

Suppose m∈N0, (σk)k∈N⊂[0,1], limk→∞σk=:σ∈[0,1], sk:=m+σk, s:=m+σ>0, and N>2⁢max⁡{s,sk} for all k∈N. Let Ω⊂RN be a smooth bounded G-invariant domain and let uk∈D0sk⁢(Ω)ϕ be such that

(3.16) C - 1 < | u k | 2 s k ⋆ < C ⁢ ⁢ for all ⁢ k ∈ ℕ ⁢ and for some ⁢ C > 1 ,

(3.17) ∥ J s k ′ ⁢ ( u k ) ∥ ( D s k ⁢ ( ℝ N ) ) ′ = o ⁢ ( 1 ) ⁢ ⁢ as ⁢ k → ∞ ,

(3.18) u k → 0 ⁢ ⁢ in ⁢ D 0 s - δ ⁢ ( Ω ) ⁢ as ⁢ k → ∞ ⁢ for some ⁢ δ ∈ ( 0 , σ ) .

Then there exist sequences (λk)k∈N⊂(0,∞), (ξk)k∈N⊂(RN)G, and a constant C1>0 such that λk→0, ξk→ξ∈(RN)G,

(3.19) dist ⁡ ( ξ k , Ω ) ≤ C 1 ⁢ λ k ,

and the rescaling

(3.20) w k ⁢ ( y ) := λ k N 2 - s ⁢ u k ⁢ ( λ k ⁢ y + ξ k ) , y ∈ ℝ N ,

satisfies that, up to a subsequence,

η ⁢ w k → η ⁢ w in ⁢ D s - δ ⁢ ( ℝ N ) ⁢ as ⁢ k → ∞ ⁢ for all ⁢ η ∈ C c ∞ ⁢ ( ℝ N ) , δ ∈ ( 0 , s ) ,

and for some w∈Ds⁢(RN)ϕ∖{0}.

Proof.

Let C>1 as in (3.16), κN,s as in (2.6), and let τ>0 be such that

(3.21) τ < min ⁡ { ( 3 ⁢ κ N , s k 2 ) - N 2 ⁢ s k , C - 1 } for all ⁢ k ∈ ℕ .

By (3.16), there are (λk)k∈ℕ⊂(0,∞) and (xk)k∈ℕ⊂ℝN such that, passing to a subsequence,

(3.22) sup x ∈ ℝ N ⁡ ∫ B λ k ⁢ ( x ) | u k | 2 s k ⋆ = ∫ B λ k ⁢ ( x k ) | u k | 2 s k ⋆ = τ .

For the chosen sequences (λk)k∈ℕ and (xk)k∈ℕ, let C0>0 and (ξk)k∈ℕ be given by Lemma 2.6. Then |gk⁢xk-ξk|≤C0⁢λk for some gk∈G and, since |uk| is G-invariant, we have that

(3.23) τ = ∫ B λ k ⁢ ( g k ⁢ x k ) | u k | 2 s k ⋆ ≤ ∫ B C 1 ⁢ λ k ⁢ ( ξ k ) | u k | 2 s k ⋆ ,

where C1:=C0+1 and, in particular, (3.19) holds.

We claim that (ξk)k∈ℕ⊂(ℝN)G. Otherwise, by Lemma 2.6, we have, for each m∈ℕ, m different elements g1,…,gm∈G such that BC1⁢λk⁢(gi⁢ξk)∩BC1⁢λk⁢(gj⁢ξk)=∅ for k large enough. Therefore, from (3.23),

m ⁢ τ ≤ ∑ i = 1 m ∫ B C 1 ⁢ λ k ⁢ ( g i ⁢ ξ k ) | u k | 2 s k ⋆ ≤ ∫ Ω | u k | 2 s k ⋆ < C for every ⁢ m ∈ ℕ ,

which yields a contradiction to (3.16). Thus, (ξk)k∈ℕ⊂(ℝN)G. Let wk be given by (3.20). Since uk is ϕ-equivariant and ξk is a G-fixed point, we have that wk is ϕ-equivariant. Observe that, by (3.22), (3.23), and a change of variables,

(3.24) τ = sup z ∈ ℝ N ⁡ ∫ B 1 ⁢ ( z ) | w k | 2 s k ⋆ ≤ ∫ B C 1 ⁢ ( 0 ) | w k | 2 s k ⋆ .

Similarly, by (3.16) and (3.17), (wk)k∈ℕ is uniformly bounded in Dsk⁢(ℝN). By Lemma 3.2, there is w∈Ds⁢(ℝN)ϕ such that, passing to a subsequence,

η ⁢ w k → η ⁢ w in ⁢ D s - ε ⁢ ( ℝ N ) ϕ ⁢ as ⁢ k → ∞ ⁢ for all ⁢ η ∈ C c ∞ ⁢ ( ℝ N ) ⁢ and ⁢ ε ∈ ( 0 , s ) .

Now, we prove by contradiction that w≠0. Assume that w=0. Given φ∈Cc∞⁢(ℝN), we set

ϑ ⁢ ( x ) := 1 μ ⁢ ( G ) ⁢ ∫ G φ 2 ⁢ ( g ⁢ x ) ⁢ d μ and ϑ k ⁢ ( x ) = ϑ ⁢ ( x - ξ k λ k ) .

Using that wk is ϕ-equivariant and according to (2.9), a direct computation yields (φ2⁢wk)ϕ=ϑ⁢wk, whence ϑk⁢uk is ϕ-equivariant and ∥ϑk⁢uk∥sk is uniformly bounded. From here, using Lemma 2.3 and (3.17),

(3.25) J s k ′ ⁢ ( w k ) ⁢ ( φ 2 ⁢ w k ) = J s k ′ ⁢ ( w k ) ⁢ ϑ ⁢ w k = J s k ′ ⁢ ( u k ) ⁢ ( ϑ k ⁢ u k ) = o ⁢ ( 1 ) as ⁢ k → ∞ .

By Lemma 3.5,

∥ w k ⁢ φ ∥ s k 2 ≤ 3 2 ⁢ ℰ s k ⁢ ( w k , φ 2 ⁢ w k ) + o ⁢ ( 1 ) as ⁢ k → ∞ .

Let φ∈Cc∞⁢(B1⁢(z)) with z∈ℝN. Then, by Hölder’s inequality and (3.25),

∥ w k ⁢ φ ∥ s k 2 ≤ 3 2 ⁢ ∫ B 1 ⁢ ( z ) | w k | 2 s k ⋆ - 2 ⁢ | w k ⁢ φ | 2 + o ⁢ ( 1 )

≤ 3 2 ⁢ ( ∫ B 1 ⁢ ( z ) | w k | 2 s k ⋆ ) 2 s k ⋆ - 2 2 s k ⋆ ⁢ ( ∫ ℝ N | φ ⁢ w k | 2 s k ⋆ ) 2 2 s k ⋆ + o ⁢ ( 1 )

≤ 3 2 ⁢ τ 2 ⁢ s k N ⁢ | φ ⁢ w k | 2 s k ⋆ 2 + o ⁢ ( 1 ) .

By Theorem 2.1 and (3.21),

(3.26) ∥ φ ⁢ w k ∥ s k 2 ≤ 3 2 ⁢ τ 2 ⁢ s k N ⁢ κ N , s 2 ⁢ ∥ φ ⁢ w k ∥ s k 2 + o ⁢ ( 1 ) ≤ 1 2 ⁢ ∥ φ ⁢ w k ∥ s k 2 + o ⁢ ( 1 ) as ⁢ k → ∞ .

By (3.26), we have that ∥φ⁢wk∥sk=o⁢(1), and therefore (by Theorem 2.1) |φ⁢wk|2sk⋆=o⁢(1) as k→∞ for any φ∈Cc∞⁢(B1⁢(z)), which contradicts (3.24). Therefore,

(3.27) w ≠ 0 in ⁢ ℝ N .

Then, passing to a subsequence, ξk→ξ∈(ℝN)G as k→∞ and, by (3.18), (3.20), and (3.27), we conclude that λk→0 as k→∞. ∎

4 A Concentration Result

In this section, we show a concentration result following the strategy from [19, Theorem 2.5] and [20, Theorem 3.5] (see also [48, Theorem 8.13]). Recall that AG denotes the set of G-fixed points of A⊂ℝN.

Theorem 4.1.

Assume that G and ϕ satisfy (A1) and (A2). Let s>0, N≥1, N>2⁢s, let Ω be a G-invariant bounded smooth domain in RN, and let uk∈D0s⁢(Ω)ϕ be such that

(4.1) J s ⁢ ( u k ) → c s ϕ ⁢ ( Ω ) 𝑎𝑛𝑑 J s ′ ⁢ ( u k ) → 0 in ⁢ ( D 0 s ⁢ ( Ω ) ϕ ) ′ ⁢ as ⁢ k → ∞ .

Then, up to a subsequence, one of the following two possibilities occurs:

( u k ) k ∈ ℕ converges strongly in D 0 s ⁢ ( Ω ) to a minimizer of J s on 𝒩 s ϕ ⁢ ( Ω ) .
There exist sequences ( ξ k ) k ∈ ℕ ⊂ ( ℝ N ) G , (λk)k∈ℕ⊂(0,∞), and a nontrivial solution w to

(4.2) ( - Δ ) s ⁢ w = | w | 2 s ⋆ - 2 ⁢ w , w ∈ D 0 s ⁢ ( 𝔼 ) ϕ ,

with the following properties:
1. λ k → 0 , ξk→ξ, ξ∈(Ω¯)G, and λk-1⁢dist⁡(ξk,Ω)→d∈[0,∞].
2. If d = ∞ , then 𝔼 = ℝ N and ξ k ∈ Ω .
3. If d ∈ [ 0 , ∞ ) , then ξ ∈ ∂ ⁡ Ω and 𝔼 = { x ∈ ℝ N : x ⋅ ν > d ¯ } , where ν is the inward-pointing unit normal to ∂⁡Ω at ξ and d¯∈{d,-d}. Moreover, 𝔼 is G-invariant, 𝔼G≠0, and ΩG≠0.
4. w ∈ 𝒩 s ϕ ⁢ ( 𝔼 ) and J s ⁢ ( w ) = c s ϕ ⁢ ( ℝ N ) .
5. One has
  
  lim k → ∞ ⁡ ∥ u k - λ k - N 2 + s ⁢ w ⁢ ( ⋅ - ξ k λ k ) ∥ s = 0 .

Proof.

Since

J s ′ ⁢ ( u k ) ⁢ u k = ∥ u k ∥ s 2 - | u k | 2 s ⋆ 2 s ⋆ ,

we have, by (4.1), that

(4.3) s N ⁢ ∥ u k ∥ s 2 = J s ⁢ ( u k ) - 1 2 s ⋆ ⁢ J s ′ ⁢ ( u k ) ⁢ u k ≤ C + o ⁢ ( 1 ) ⁢ ∥ u k ∥ s .

Therefore, (uk)k∈ℕ is bounded in D0s⁢(Ω)ϕ and, up to a subsequence, there exists u∈D0s⁢(Ω)ϕ such that

(4.4) u k ⇀ u weakly in ⁢ D 0 s ⁢ ( Ω ) ϕ ⁢ as ⁢ k → ∞ .

By Theorem 2.2, up to a subsequence, uk→u strongly in Lν⁢(Ω) for any ν∈[2,2s⋆). Then

lim k → ∞ ⁡ ∫ Ω | u k | 2 s ⋆ - 2 ⁢ u k ⁢ φ = ∫ Ω | u | 2 s ⋆ - 2 ⁢ u ⁢ φ for all ⁢ φ ∈ C c ∞ ⁢ ( Ω ) ϕ .

Hence, for any φ∈Cc∞⁢(Ω)ϕ,

J s ′ ⁢ ( u ) ⁢ φ = ℰ s ⁢ ( u , φ ) - ∫ Ω | u ⁢ ( x ) | 2 s ⋆ - 2 ⁢ u ⁢ ( x ) ⁢ φ ⁢ ( x ) ⁢ d x

= lim k → ∞ ⁡ [ ℰ s ⁢ ( u k , φ ) - ∫ Ω | u k ⁢ ( x ) | 2 s ⋆ - 2 ⁢ u k ⁢ ( x ) ⁢ φ ⁢ ( x ) ⁢ d x ]

(4.5) = lim k → ∞ ⁡ J s ′ ⁢ ( u k ) ⁢ φ = 0 .

We now consider two cases.

(I) If u≠0, then u∈𝒩sϕ⁢(Ω) and, by (4.1) and (4.3),

c s ϕ ⁢ ( Ω ) ≤ J s ⁢ ( u ) = 1 2 ⁢ ∥ u ∥ s 2 - 1 2 s ⋆ ⁢ | u | 2 s ⋆ 2 s ⋆ = s N ⁢ ∥ u ∥ s 2 ≤ lim inf k → ∞ ⁡ s N ⁢ ∥ u k ∥ s 2 = c s ϕ ⁢ ( Ω ) + o ⁢ ( 1 ) .

This together with (4.4) implies that uk→u strongly in D0s⁢(Ω)ϕ, and thus Js⁢(u)=csϕ⁢(Ω).

(II) If u=0, then (4.1) and (4.3) imply that

∫ Ω | u k ⁢ ( x ) | 2 s ⋆ ⁢ d x = N s ⁢ ( J s ⁢ ( u k ) - 1 2 ⁢ J s ′ ⁢ ( u k ) ⁢ u k ) → N s ⁢ c s ϕ ⁢ ( Ω ) as ⁢ k → ∞ .

Note that uk satisfies the assumptions of Lemma 3.6 (with sk=s for all k∈ℕ). Let λk→0, ξk→ξ, wk, and w be as given by Lemma 3.6 and define

d := lim k → ∞ ⁡ λ k - 1 ⁢ dist ⁡ ( ξ k , ∂ ⁡ Ω ) ∈ [ 0 , ∞ ] , Ω k := { y ∈ ℝ N : λ k ⁢ y + ξ k ∈ Ω } .

If d=∞, then, by (3.19), we have that ξk∈Ω. Hence, for every X⊂⊂ℝN, there exists k0 such that X⊂Ωk for all k≥k0. Thus, for d=∞ we set 𝔼:=ℝN. Otherwise, if d∈[0,∞), then, as λk→0, we have that ξ∈∂⁡Ω. If a subsequence of (ξk) is contained in Ω¯, then we set d¯:=-d, otherwise we take d¯:=d. We define ℍ:={y∈ℝN:y⋅ν>d¯}, where ν is the inward-pointing unit normal to ∂⁡Ω at ξ. Since ξ is a G-fixed point, so is ν. Thus ΩG≠∅, ℍ is G-invariant, and ℍG≠∅. If X is compact and X⊂ℍ, then there exists k0 such that X⊂Ωk for all k≥k0. Moreover, if X is compact and X⊂ℝN∖ℍ¯, then X⊂ℝN∖Ωk for k large enough. As wk→w a.e. in ℝN, this implies that w=0 a.e. in ℝN∖ℍ. So w∈D0s⁢(ℍ)ϕ. Then, for d<∞, we set 𝔼:=ℍ.

For φ∈Cc∞⁢(𝔼)ϕ, we define

φ k ⁢ ( x ) := λ k - N 2 + s ⁢ φ ⁢ ( x - ξ k λ k ) .

Since ξk is a G-fixed point, φk is ϕ-equivariant and there is k0 large enough such that supp⁡φk⊂Ω. By (1.2), φk is uniformly bounded in D0s⁢(Ω). Hence, (4.5), Lemmas 2.3 and 3.6, and a direct computation yield Js′⁢(wk)⁢φ=Js′⁢(uk)⁢φk=o⁢(1) as k→∞. Therefore, w is a nontrivial weak solution of (4.2). From Lemma 2.5, we conclude that csϕ⁢(Ω)=csϕ⁢(𝔼)=csϕ⁢(ℝN). Hence,

c s ϕ ⁢ ( ℝ N ) ≤ J s ⁢ ( w ) = s N ⁢ ∥ w ∥ s 2 ≤ lim inf k → ∞ ⁡ s N ⁢ ∥ w k ∥ s 2 = s N ⁢ lim inf k → ∞ ⁡ ∥ u k ∥ s 2 = c s ϕ ⁢ ( ℝ N ) .

Thus, Js⁢(w)=c∞ϕ and wk→w strongly in Ds⁢(ℝN) as k→∞. By a change of variable, this implies that

o ⁢ ( 1 ) = ∥ w k - w ∥ s = ∥ u k - λ k - N 2 + s ⁢ w ⁢ ( ⋅ - ξ k λ k ) ∥ s as ⁢ k → ∞ .

This ends the proof. ∎

5 Existence, Nonexistence, and Convergence of Solutions in Symmetric Bounded Domains

We begin this section with the proof of the nonexistence result stated in the introduction.

Proof of Proposition 1.2.

By contradiction, let u be a nontrivial nonnegative solution of (1.7). Let

𝒟 s ⁢ u ⁢ ( z ) := lim x → z x ∈ B , x | x | = z ⁡ u ⁢ ( x ) ( 1 - | x | 2 ) s = 1 2 s ⁢ lim x → z x ∈ B , x | x | = z ⁡ u ⁢ ( x ) ( 1 - | x | ) s = 1 2 s ⁢ lim x → z x ∈ B , x | x | = z ⁡ u ⁢ ( x ) dist ( x , ℝ N ∖ B ) s .

By [3, Corollary 1.9 and Lemma 2.1], we have that

𝒟 s ⁢ u ⁢ ( z ) = Γ ⁢ ( N 2 ) 4 s ⁢ π N 2 ⁢ Γ ⁢ ( s ) 2 ⁢ s ⁢ ∫ B ( 1 - | y | 2 ) s | y - z | N ⁢ u ⁢ ( y ) 2 s ⋆ - 1 ⁢ d y > 0 for ⁢ z ∈ ∂ ⁡ B .

However, by the Pohozaev identity [44, Corollary 1.7],

∫ ∂ ⁡ B | 𝒟 s ⁢ u ⁢ ( z ) | 2 ⁢ 𝑑 σ = 0 .

This yields a contradiction, and therefore (1.7) has no nontrivial nonnegative solutions. ∎

For bounded domains without fixed points, we have the following existence result.

Proposition 5.1.

Assume that G and ϕ verify assumptions (A1) and (A2). Let Ω be a G-invariant bounded smooth domain in RN such that Ω¯G=∅ and let s>0. Then the problem

(5.1) { ( - Δ ) s ⁢ u = | u | 2 s ⋆ - 2 ⁢ u , u ∈ D 0 s ⁢ ( Ω ) ϕ ,

has a least-energy solution. The solution is sign-changing if ϕ:G→Z2 is surjective.

Proof.

By Lemma 2.4 (i) and (iii) and [48, Theorem 2.9], there is a sequence (uk) such that (4.1) holds. Then, by Theorem 4.1, since alternative (ii) cannot hold due to the lack of fixed points in Ω¯, we conclude that Js attains a minimum u∈𝒩sϕ⁢(Ω). Then there is a Lagrange multiplier λ∈ℝ such that

(5.2) J s ′ ⁢ ( u ) ⁢ φ = λ ⁢ ( 2 ⁢ ℰ s ⁢ ( u , φ ) - 2 s ⋆ ⁢ ∫ Ω | u | 2 s ⋆ - 2 ⁢ u ⁢ φ ) for all ⁢ φ ∈ D 0 s ⁢ ( Ω ) ϕ .

Testing with φ=u, we obtain that

0 = ( 1 - 2 ⁢ λ ) ⁢ ∥ u ∥ s 2 + ( 2 s ⋆ ⁢ λ - 1 ) ⁢ | u | 2 s ⋆ 2 s ⋆ = ( 1 - 2 ⁢ λ + 2 s ⋆ ⁢ λ - 1 ) ⁢ | u | 2 s ⋆ 2 s ⋆ = ( 2 s ⋆ - 2 ) ⁢ λ ⁢ | u | 2 s ⋆ 2 s ⋆ .

Since u≠0, this implies that λ=0. Then (5.2) and Lemma 2.3 imply that u is a weak solution of (5.1). ∎

Next, we show some convergence properties of the solutions to (5.1). We begin with an auxiliary lemma.

Lemma 5.2.

Let G and ϕ verify (A1) and (A2) and let Ω be a G-invariant bounded smooth domain in RN such that Ω¯G=∅. Let s>0, N>2⁢s, and 0<δ<min⁡{s,N2-s}. For each t∈(s-δ,s+δ), let ut be a least-energy solution to (5.1) given by Proposition 5.1. Then there is a constant C>1 depending only on δ, Ω, and s such that

(5.3) C - 1 < ∥ u t ∥ t < C for all ⁢ t ∈ ( s - δ , s + δ ) .

Proof.

Let N, s, δ, t, and ut be as in the statement and let φ∈Cc∞⁢(Ω)∖{0}. Then

k t ⁢ φ ∈ 𝒩 t , k t := ( ∥ φ ∥ t 2 | φ | 2 t ⋆ 2 t ⋆ ) 1 2 t ⋆ - 2 .

Then, since 2t⋆-2>0 for all t∈[s-δ,s+δ],

c t ϕ ( Ω ) ≤ J t ( k t φ ) ≤ sup t ∈ ( s - δ , s + δ ) J t ( k t φ ) = : C 1 ,

where C1>0 depends only on φ, s, δ, and Ω. Moreover, since ut∈𝒩tϕ is a least-energy solution,

∥ u t ∥ t 2 = N t c t ϕ ( Ω ) ≤ N s - δ C 1 = : C 2 .

This establishes the upper bound in (5.3). To obtain the lower bound, let

F t ⁢ ( u ) := ∥ u ∥ t 2 - | u | 2 t ⋆ 2 t ⋆ ≥ ∥ u ∥ t 2 - κ N , t 2 t ⋆ ⁢ ∥ u ∥ t 2 t ⋆ for ⁢ u ∈ D 0 t ⁢ ( Ω ) ,

where κN,t is explicitly given by (2.6). In particular, by the definition of δ, we have N>2⁢t, and therefore

sup t ∈ ( s - δ , s + δ ) κ N , t = sup t ∈ ( s - δ , s + δ ) 2 - 2 ⁢ t π - t Γ ⁢ ( N - 2 ⁢ t 2 ) Γ ⁢ ( N + 2 ⁢ t 2 ) ( Γ ⁢ ( N ) Γ ⁢ ( N 2 ) ) 2 ⁢ t N = : K ,

where K depends only on N,s, and δ. Then, for ∥u∥t<1,

F t ⁢ ( u ) ≥ ∥ u ∥ t 2 ⁢ ( 1 - K 2 t ⋆ ⁢ ∥ u ∥ t 2 t ⋆ - 2 ) ≥ ∥ u ∥ t 2 ⁢ ( 1 - ( K + 1 ) 2 s + δ ⋆ ⁢ ∥ u ∥ t 4 ⁢ ( s - δ ) N - 2 ⁢ ( s - δ ) ) .

In particular, Ft⁢(u)>0 for all t∈(s-δ,s+δ) if

0 < ∥ u ∥ t < ( K + 1 ) - N ⁢ ( 2 ⁢ δ + N - 2 ⁢ s ) 2 ⁢ ( s - δ ) ⁢ ( N - 2 ⁢ ( δ + s ) ) = : a .

Since Ft⁢(ut)=0 because ut∈𝒩t, necessarily ∥ut∥t>a for t∈(s-δ,s+δ). This yields the lower bound in (5.3). ∎

Our main convergence result is the following.

Theorem 5.3.

Let G and ϕ verify (A1) and (A2), let Ω be a G-invariant bounded smooth domain in RN such that Ω¯G=∅, and let m∈N0, (σk)k∈N⊂[0,1], limk→∞σk=:σ∈[0,1], sk:=m+σk>0, s:=m+σ>0, and N>2⁢max⁡{s,sk} for all k∈N. Let usk be a least-energy solution of

( - Δ ) s k ⁢ u s k = | u s k | 2 s k ⋆ - 2 ⁢ u s k , u s k ∈ D 0 s k ⁢ ( Ω ) ϕ .

Then, up to a subsequence,

u s k → u strongly in ⁢ D 0 s - δ ⁢ ( Ω ) ⁢ as ⁢ k → ∞ ⁢ for all ⁢ δ ∈ ( 0 , s ) ,

where u is a least-energy solution of

(5.4) ( - Δ ) s ⁢ u = | u | 2 s ⋆ - 2 ⁢ u , u ∈ D 0 s ⁢ ( Ω ) ϕ .

Proof.

Let s, sk, and usk be as in the assumptions and let

0 < δ < min ⁡ { s , N 2 - s , ( N - 2 ⁢ s ) 2 2 ⁢ ( N + 2 ⁢ s ) } .

In the following, C>1 denotes possibly different constants depending at most on s, δ, N, and Ω. Passing to a subsequence, we may assume that sk∈(s-δ,s+δ) for all k∈ℕ. By Lemma 5.2,

C - 1 < ∥ u s k ∥ s k 2 = | u s k | 2 s k ⋆ 2 s k ⋆ < C for all ⁢ k ∈ ℕ .

By Lemma 3.1,

∥ u s k ∥ s - δ 2 2 ≤ C ⁢ ∥ u s k ∥ s k 2 < C .

Then, by Theorem 2.2, there is u∈D0s-δ⁢(Ω) such that

(5.5) u s k → u ⁢ in ⁢ D 0 s - δ ⁢ ( Ω ) , u s k → u ⁢ in ⁢ L p ⁢ ( Ω ) ⁢ for ⁢ p ∈ [ 2 , 2 s - δ ⋆ ) as ⁢ k → ∞ .

Note that, since

δ < ( N - 2 ⁢ s ) 2 2 ⁢ ( N + 2 ⁢ s ) ,

we have

(5.6) 2 s ⋆ - 1 < 2 s - δ ⋆ .

Moreover, using Fatou’s Lemma as in (3.5), we have that ∥u∥s2≤lim infk→∞⁡∥usk∥sk2<C, and therefore u∈D0s⁢(Ω). Since usk is a least-energy solution, we have from integration by parts (see, e.g., [5, Lemma 1.5]) that

0 = J s k ′ ⁢ ( u s k ) ⁢ φ = ℰ s k ⁢ ( u s k , φ ) - ∫ Ω | s k | 2 s k ⋆ - 2 ⁢ u s k ⁢ φ = ∫ Ω u s k ⁢ ( - Δ ) s k ⁢ φ - ∫ Ω | u s k | 2 s k ⋆ - 2 ⁢ u s k ⁢ φ .

Note that, by (5.6),

2 s k ⋆ - 1 = 2 s ⋆ - 1 + o ⁢ ( 1 ) < 2 s - δ ⋆ - ε + o ⁢ ( 1 ) < 2 s - δ ⋆ as ⁢ k → ∞ ,

with

ε = 1 2 ⁢ ( 2 s - δ ⋆ - 2 s ⋆ + 1 ) > 0 .

Then, by (5.5) and Lemma A.2,

0 = ∫ Ω u ⁢ ( - Δ ) s ⁢ φ - ∫ Ω | u | 2 s ⋆ - 2 ⁢ u ⁢ φ for all ⁢ φ ∈ C c ∞ ⁢ ( Ω ) ,

that is, u is a weak solution of the limit problem (5.4). Note that

(5.7) u ≠ 0 in ⁢ Ω .

Indeed, assume by contradiction that u=0. Then usk satisfies the assumptions of Lemma 3.6. Let λk→0 and ξk→ξ be given by Lemma 3.6 and define

d := lim k → ∞ ⁡ λ k - 1 ⁢ dist ⁡ ( ξ k , ∂ ⁡ Ω ) ∈ [ 0 , ∞ ] .

If d=∞, then, by (3.19), ξk∈Ω. But this cannot happen since Ω¯G=∅ and ξk∈(ℝN)G. On the other hand, if d∈[0,∞), then, as λk→0, we have that ξ∈∂⁡Ω, which also cannot happen because Ω¯G=∅. We have reached a contradiction and (5.7) follows.

Next, we show that u is a least-energy solution, namely that Js⁢(u)=csϕ⁢(Ω). By Lemma 5.2, there is C>0 such that C-1<cskϕ<C for all k∈ℕ. In particular, passing to a subsequence, there is c* such that cskϕ→c* as k→∞. Then, using Fatou’s Lemma as in (3.5), we obtain

c s ϕ ≤ J s ⁢ ( u )

= ( 1 2 - 1 2 s ⋆ ) ⁢ ∥ u ∥ s 2

≤ lim inf k → ∞ ⁡ ( 1 2 - 1 2 s k ⋆ ) ⁢ ∥ u s k ∥ s k 2

= lim inf k → ∞ ⁡ J s k ⁢ ( u s k )

(5.8) = lim inf k → ∞ ⁡ c s k ϕ = c * .

On the other hand, by Proposition 5.1, there is us∈𝒩s such that Js⁢(us)=csϕ. Then

(5.9) t k := ( ∥ u s ∥ s k 2 | u s | 2 s k ⋆ 2 s k ⋆ ) 1 2 s k ⋆ - 2 = 1 + o ⁢ ( 1 ) as ⁢ k → ∞ ,

and

(5.10) ∥ u s ∥ s k = ∥ u s ∥ s + o ⁢ ( 1 ) , | u s | 2 s k ⋆ = | u s | 2 s ⋆ + o ⁢ ( 1 ) as ⁢ k → ∞ .

But then, using the minimality of usk and using (5.9) and (5.10), we obtain

c * ϕ + o ⁢ ( 1 ) = c s k ϕ = J s k ⁢ ( u s k ) ≤ J s k ⁢ ( t k ⁢ u s ) = J s ⁢ ( u s ) + o ⁢ ( 1 ) = c s ϕ + o ⁢ ( 1 ) as ⁢ k → ∞ .

Therefore, c*ϕ≤csϕ and, with (5.8), we conclude that c*ϕ=csϕ and that Js⁢(u)=csϕ. ∎

Proof of Theorem 1.1.

The first part (existence) follows from Proposition 5.1, and the second part (convergence) follows from Theorem 5.3. ∎

6 Existence and Convergence of Entire Solutions

Recall the definition of 𝒩sϕ given in (2.14) and of Js given in (2.8). We begin with an existence theorem.

Theorem 6.1.

Let N∈N, s>0, N>2⁢s, let G be a closed subgroup of O⁢(N), and let ϕ:G→Z2 be a continuous homomorphism satisfying (A1) and (A2). Then Js attains its minimum on Nsϕ⁢(RN). Consequently, the problem

(6.1) ( - Δ ) s ⁢ u = | u | 2 s ⋆ - 2 ⁢ u , u ∈ D s ⁢ ( ℝ N ) ϕ ,

has a nontrivial ϕ-equivariant solution. The solution is sign-changing if ϕ is surjective.

Proof.

The unitary ball B={x∈ℝN:|x|<1} is G-invariant for every subgroup G of O⁢(N). Since 0∈BG, we have csϕ⁢(B)=csϕ⁢(ℝN) by Lemma 2.5. By Lemma 2.4 (i) and (iii) and by [48, Theorem 2.9], we obtain the existence of a sequence (uk)k∈ℕ⊂D0s⁢(B)ϕ such that Js⁢(uk)→csϕ⁢(Ω) and Js′⁢(uk)→0 in (D0s⁢(B)ϕ)′ as k→∞. Then, by Theorem 4.1, there exists

u ∈ 𝒩 s ϕ ⁢ ( 𝔼 ) ⊂ 𝒩 s ϕ ⁢ ( ℝ N ) with ⁢ J s ⁢ ( u ) = c s ϕ ⁢ ( ℝ N ) ,

and therefore Js attains its minimum on 𝒩sϕ⁢(ℝ). Arguing as in Proposition 5.1, we conclude that u is a weak solution of (6.1). ∎

Next, we show some convergence properties of the solutions to (1.1) as sk→s, where s>0. We begin with an auxiliary lemma.

Lemma 6.2.

Assume the hypothesis of Theorem 6.1. Let s>0, N>2⁢s, and 0<δ≤min⁡{s,N2-s}. For each t∈(s-δ,s+δ), let ut be a least-energy solution to (6.1) given by Theorem 6.1. Then there is a constant C>1 depending only on δ and s such that

C - 1 < ∥ u t ∥ t < C for all ⁢ t ∈ ( s - δ , s + δ ) .

Proof.

Repeat the proof from Lemma 5.2 with Ω=ℝN. ∎

Theorem 6.3.

Assume that G and ϕ verify assumptions (A1) and (A2). Let N∈N, let (sk)k∈N⊂(0,∞) be such that sk→s=m+σ>0 as k→∞ with m∈N0, σ∈[0,1], and N>2⁢max⁡{s,sk} for all k∈N. For κN,s as in (2.6) and τ>0 such that

(6.2) τ < ( 3 ⁢ κ N , s k 2 ) - N 2 ⁢ s k for all ⁢ k ∈ ℕ ,

let wsk∈Dsk⁢(RN)ϕ be a least-energy solution of (-Δ)sk⁢wsk=|wsk|2sk⋆-2⁢wsk satisfying that

(6.3) ∫ B 1 ⁢ ( 0 ) | w s k | 2 s k ⋆ = τ for all ⁢ k ∈ ℕ .

Then there is a least-energy solution w∈Ds⁢(RN)ϕ of (-Δ)s⁢w=|w|2s⋆-2⁢w such that, up to a subsequence,

(6.4) η ⁢ w s k → η ⁢ w in ⁢ D s - δ ⁢ ( ℝ N ) ⁢ as ⁢ k → ∞ ⁢ for all ⁢ η ∈ C c ∞ ⁢ ( ℝ N ) ⁢ and ⁢ δ ∈ ( 0 , σ ) .

Proof.

Let sk and wsk as in the statement. In the following, C>0 denotes possibly different constants independent of k. By Lemma 6.2, there is C>0 such that

(6.5) C - 1 < c s k ϕ ⁢ ( ℝ N ) < C for all ⁢ k ∈ ℕ .

We split the proof in steps.

Step 1: Find a limit profile for wsk. Let ζ∈Cc∞⁢(ℝ) be such that

(6.6) 0 ≤ ζ ≤ 1 in ⁢ ℝ , ζ ⁢ ( r ) = 1 if ⁢ | r | ≤ 1 , ζ ⁢ ( r ) = 0 if ⁢ | r | ≥ 2 ,

and let

(6.7) w s k n ⁢ ( x ) := w s k ⁢ ( x ) ⁢ ζ n ⁢ ( x ) , ζ n ⁢ ( x ) := ζ ⁢ ( | x | n ) for ⁢ n ∈ ℕ ⁢ and ⁢ x ∈ ℝ N .

By Lemma A.4 and the triangle inequality,

(6.8) ∥ w s k n ∥ s k < C for all ⁢ n , k ∈ ℕ .

By Lemma 3.2, there is wsn∈Ds⁢(ℝN)ϕ such that, up to a subsequence,

(6.9) φ ⁢ w s k n → φ ⁢ w s n in ⁢ D s - δ ⁢ ( ℝ N ) ⁢ as ⁢ k → ∞ ⁢ for all ⁢ n ∈ ℕ ⁢ and ⁢ φ ∈ C c ∞ ⁢ ( ℝ N ) .

By a standard diagonalization argument, we may assume that wsn=wsm in Bn⁢(0) for all m>n, m,n∈ℕ. Moreover, using Fatou’s Lemma (as in (3.5)) and (6.8), we obtain

∥ w s n ∥ s ≤ lim inf k → ∞ ⁡ ∥ w s k n ∥ s k < C for all ⁢ n ∈ ℕ .

Therefore, there is w∈Ds⁢(ℝN)ϕ such that, up to a subsequence,

w s n ⇀ w weakly in ⁢ D s ⁢ ( ℝ N ) ⁢ as ⁢ n → ∞ .

But then wsn=w in Bn⁢(0) for all n∈ℕ. By Lemma 3.2, we deduce that

(6.10) w s k → w in ⁢ L loc q ⁢ ( ℝ N ) ⁢ as ⁢ k → ∞ ⁢ for ⁢ q ∈ [ 1 , 2 s ⋆ ) ,

and (6.4) follows from (6.9) taking n large enough.

Step 2: Show that w is a weak solution. Let φ∈Cc∞⁢(ℝN) and n∈ℕ. Observe that, by (6.10) and Lemma A.2,

(6.11) lim k → ∞ ⁡ ∫ B n ⁢ ( 0 ) w s k ⁢ ( - Δ ) s k ⁢ φ = ∫ B n ⁢ ( 0 ) w ⁢ ( - Δ ) s ⁢ φ = ∫ ℝ N w ⁢ ( - Δ ) s ⁢ φ

and, by Hölder’s inequality, (6.8), Lemma 2.1, and Lemma A.2,

lim k → ∞ ⁡ ∫ ℝ N ∖ B n ⁢ ( 0 ) w s k ⁢ ( - Δ ) s k ⁢ φ ≤ lim k → ∞ ⁡ | w s k | 2 s k ⋆ ⁢ ( ∫ ℝ N ∖ B n ⁢ ( 0 ) | ( - Δ ) s k ⁢ φ | ( 2 s k ⋆ ) ′ ) 1 ( 2 s k ⋆ ) ′

≤ C ⁢ lim k → ∞ ⁡ ( ∫ ℝ N ∖ B n ⁢ ( 0 ) | ( - Δ ) s k ⁢ φ | 2 ⁢ N N + 2 ⁢ s k ) N + 2 ⁢ s k 2 ⁢ N

(6.12) = C ⁢ ( ∫ ℝ N ∖ B n ⁢ ( 0 ) | ( - Δ ) s ⁢ φ | 2 ⁢ N N + 2 ⁢ s ) N + 2 ⁢ s 2 ⁢ N = 0 ,

where

( - Δ ) s ⁢ φ ∈ L 2 ⁢ N N + 2 ⁢ s ⁢ ( ℝ N )

by Lemma A.1. Then (6.11), (6.12), and integration by parts (see, e.g., [5, Lemma 1.5]) imply that

lim k → ∞ ⁡ ℰ s k ⁢ ( w s k , φ ) = lim n → ∞ ⁡ lim k → ∞ ⁡ ∫ B n ⁢ ( 0 ) w s k ⁢ ( - Δ ) s k ⁢ φ + ∫ ℝ N ∖ B n ⁢ ( 0 ) w s k ⁢ ( - Δ ) s k ⁢ φ = ∫ ℝ N w ⁢ ( - Δ ) s ⁢ φ = ℰ s ⁢ ( w , φ ) .

Therefore, by (6.10),

(6.13) 0 = lim k → ∞ ⁡ J s k ′ ⁢ ( w s k ) ⁢ φ = ℰ s ⁢ ( w , φ ) - ∫ ℝ N | w | 2 s ⋆ - 2 ⁢ w s ⁢ φ = J s ′ ⁢ ( w ) ⁢ φ ,

and w is a weak solution of the limit problem.

Step 3: Verify that

(6.14) w ≠ 0 .

Assume, by contradiction, that w=0 and let φ∈Cc∞⁢(B1⁢(0)). Then, since wsk is a weak solution and φ2⁢wsk∈Dsk⁢(ℝN) (by Lemma A.3),

(6.15) J s k ′ ⁢ ( w s k ) ⁢ ( φ 2 ⁢ w s k ) = 0 for all ⁢ k ∈ ℕ .

Then, by (6.9) and Lemma 3.5,

∥ w s k ⁢ φ ∥ s k 2 ≤ 3 2 ⁢ ℰ s k ⁢ ( w s k , φ 2 ⁢ w s k ) + o ⁢ ( 1 ) as ⁢ k → ∞ .

Therefore, by Hölder’s inequality, (6.2), (6.3), (6.15), and the fact that supp⁡(φ)⊂B1⁢(0),

∥ w s k ⁢ φ ∥ s k 2 ≤ 3 2 ⁢ ∫ B 1 ⁢ ( 0 ) | w s k | 2 s k ⋆ - 2 ⁢ | w s k ⁢ φ | 2 + o ⁢ ( 1 )

≤ 3 2 ⁢ ( ∫ B 1 ⁢ ( 0 ) | w s k | 2 s k ⋆ ) 2 s k ⋆ - 2 2 s k ⋆ ⁢ ( ∫ ℝ N | φ ⁢ w s k | 2 s k ⋆ ) 2 2 s k ⋆ + o ⁢ ( 1 )

≤ 3 2 ⁢ τ 2 ⁢ s k N ⁢ | φ ⁢ w s k | 2 s k ⋆ 2 + o ⁢ ( 1 ) as ⁢ k → ∞ .

Using Theorem 2.1 and (6.2), we have that

(6.16) ∥ φ ⁢ w s k ∥ s k 2 ≤ 3 2 ⁢ τ 2 ⁢ s k N ⁢ κ N , s k 2 ⁢ ∥ φ ⁢ w s k ∥ s k 2 + o ⁢ ( 1 ) ≤ 1 2 ⁢ ∥ φ ⁢ w s k ∥ s k 2 + o ⁢ ( 1 ) .

Then, by (6.16), we obtain ∥φ⁢wsk∥sk=o⁢(1). Therefore, by Theorem 2.1, |φ⁢wsk|2sk⋆=o⁢(1) as k→∞ for any φ∈Cc∞⁢(B1⁢(z)), which contradicts (6.3). Therefore, (6.14) holds.

Step 4: Show that w is a ϕ-equivariant least-energy solution. Observe that, by (6.5), there is c* such that cskϕ⁢(ℝN)→c* as k→∞ up to a subsequence. Moreover, by (6.13) and (6.14), we have that w∈𝒩s. By (6.10), we have that |wk|2sk⋆→|w|2s⋆ pointwisely in ℝN as k→∞. Then, by Fatou’s Lemma,

( 1 2 - 1 2 s ⋆ ) ⁢ | w | 2 s ⋆ 2 s ⋆ ≤ lim inf k → ∞ ⁡ ( 1 2 - 1 2 s k ⋆ ) ⁢ | w k | 2 s k ⋆ 2 s k ⋆ = lim inf k → ∞ ⁡ c s k ϕ ≤ c * .

Thus, by minimality,

(6.17) c s ϕ ⁢ ( ℝ N ) ≤ J s ⁢ ( w ) = ( 1 2 - 1 2 s ⋆ ) ⁢ | w | 2 s ⋆ 2 s ⋆ ≤ c * .

On the other hand, by Theorem 6.1, there is a ϕ-equivariant least-energy solution us∈𝒩s such that

J s ⁢ ( u s ) = c s ϕ ⁢ ( ℝ N ) .

By density, there is a sequence (us,n)n∈ℕ⊂Cc∞⁢(ℝN) such that us,n→us in Ds⁢(ℝN) as n→∞. Let

(6.18) t k , n := ( ∥ u s , n ∥ s k 2 | u s , n | 2 s k ⋆ 2 s k ⋆ ) 1 2 s k ⋆ - 2 .

Then limn→∞⁡limk→∞⁡tk,n=1. Moreover,

(6.19) lim n → ∞ ⁡ lim k → ∞ ⁡ ∥ u s , n ∥ s k = ∥ u s ∥ s , lim n → ∞ ⁡ lim k → ∞ ⁡ | u s , n | 2 s k ⋆ = | u s | 2 s ⋆ .

In particular,

lim n → ∞ ⁡ lim k → ∞ ⁡ J s k ⁢ ( t k , n ⁢ u s , n ) = J s ⁢ ( u s ) = c s ϕ .

But then, using the minimality of wsk and using (6.18) and (6.19),

c * ϕ + o ⁢ ( 1 ) = c s k ϕ = J s k ⁢ ( w s k ) ≤ J s k ⁢ ( t k , n ⁢ u s , n ) as ⁢ k → ∞ .

Therefore,

c * ϕ ≤ lim n → ∞ ⁡ lim k → ∞ ⁡ J s k ⁢ ( t k , n ⁢ u s , n ) = J s ⁢ ( u s ) = c s ϕ ⁢ ( ℝ N ) .

Together with (6.17), we conclude that c*ϕ⁢(ℝN)=csϕ⁢(ℝN) and that Js⁢(w)=csϕ⁢(ℝN). This establishes that w is a ϕ-equivariant least-energy solution of the limiting problem, and ends the proof. ∎

Proof of Theorem 1.3.

The first part (existence) follows from Theorem 6.1 and the second part (convergence) follows from Theorem 6.3. Observe that, due to the scaling invariance (1.2), an arbitrary function w∈D0sk⁢(ℝN)ϕ has a rescaling w~∈D0sk⁢(ℝN)ϕ satisfying (6.3). ∎

Communicated by Silvia Cingolani

Funding source: Dirección General de Asuntos del Personal Académico, Universidad Nacional Autónoma de México

Award Identifier / Grant number: IA101721

Funding statement: V. Hernández-Santamaría is supported by the program “Estancias posdoctorales por México” of CONACyT, Mexico. A. Saldaña is supported by UNAM-DGAPA-PAPIIT grant IA101721, Mexico.

A Auxiliary Lemmas

A.1 Convergence of Test Functions

In this subsection, we show a uniform bound for |(-Δ)sk⁢φ| whenever sk→s>0 and φ∈Cc∞⁢(ℝN) and show the convergence of ((-Δ)sk⁢φ)k∈ℕ in Lp⁢(ℝN) for any p≥1. For s>0 and k∈ℕ, set

S t n := { ψ ∈ C n ⁢ ( ℝ N ) : sup x ∈ ℝ N ⁡ ( 1 + | x | N + 2 ⁢ t ) ⁢ ∑ | α | ≤ n | ∂ α ⁡ ψ ⁢ ( x ) | < ∞ }

endowed with the norm

∥ ψ ∥ n , t := sup x ∈ ℝ N ⁡ ( 1 + | x | N + 2 ⁢ t ) ⁢ ∑ | α | ≤ n | ∂ α ⁡ ψ ⁢ ( x ) | .

The next lemma is a version of [2, Lemma B.5] with uniform estimates.

Lemma A.1.

Let m∈N0, (σk)k∈N⊂[0,1], limk→∞σk=:σ∈[0,1], sk:=m+σk>0, s:=m+σ>0, δ∈(0,s), and φ∈Cc∞⁢(RN). Then there is C=C⁢(N,s,δ)>0 such that, passing to a subsequence,

| ( - Δ ) s k ⁢ φ ⁢ ( x ) | ≤ C ⁢ ∥ φ ∥ 2 ⁢ m + 2 , s - δ 1 + | x | N + 2 ⁢ ( s - δ ) for all ⁢ x ∈ ℝ N ⁢ and ⁢ k ∈ ℕ .

Proof.

It suffices to consider σk∈(0,1). Note that (-Δ)m+σk⁢φ=(-Δ)σk⁢(-Δ)m⁢φ (this follows by Fourier transform, see also [5, Theorems 1.2 and 1.9] for a proof via direct calculations). Let ψ:=(-Δ)m⁢φ, B:=B1⁢(0), and δ∈(0,s). In the following, C>0 denotes possibly different constants depending at most on N, s, and δ.

By (2.1),

c N , σ k ≤ C ⁢ σ k ⁢ ( 1 - σ k ) for all ⁢ k ∈ ℕ .

For x∈ℝN, we have, by the mean value theorem (see [2, Lemma B.1]),

≤ C ⁢ σ k ⁢ ( 1 - σ k ) ⁢ ( | ∫ B 2 ⁢ ψ ⁢ ( x ) - ψ ⁢ ( x + y ) - ψ ⁢ ( x - y ) | y | N + 2 ⁢ σ k ⁢ d y | + | ∫ ℝ N ∖ B 2 ⁢ ψ ⁢ ( x ) - ψ ⁢ ( x + y ) - ψ ⁢ ( x - y ) | y | N + 2 ⁢ σ k ⁢ d y | )

(A.1) = : f 1 , k + f 2 , k ,

where Hψ denotes the Hessian of ψ. Let t:=s-δ and note that

(A.2) f 1 , k ≤ C ⁢ σ k ⁢ ( 1 - σ k ) ⁢ ∥ φ ∥ 2 ⁢ m + 2 , t ⁢ ∫ B ∫ 0 1 ∫ 0 1 | y | - N - 2 ⁢ σ k + 2 1 + | x + ( t - τ ) ⁢ y | N + 2 ⁢ t ⁢ d τ ⁢ d t ⁢ d y ≤ C ⁢ σ k ⁢ ∥ φ ∥ 2 ⁢ m + 2 , t 1 + | x | N + 2 ⁢ t ,

f 2 , k ≤ 2 ⁢ σ k ⁢ ( 1 - σ k ) ⁢ ( ∫ ℝ N ∖ B | ψ ⁢ ( x ) | | y | N + 2 ⁢ σ k ⁢ d y + | ∫ ℝ N ∖ B ψ ⁢ ( x + y ) | y | N + 2 ⁢ σ k ⁢ d y | )

(A.3) ≤ C ⁢ σ k ⁢ ( 1 - σ k ) ⁢ ( 1 σ k ⁢ ∥ φ ∥ 2 ⁢ m + 2 , t 1 + | x | N + 2 ⁢ t + | ∫ ℝ N ∖ B ψ ⁢ ( x + y ) | y | N + 2 ⁢ σ k ⁢ d y | ) .

Using integration by parts m times, we obtain

(A.4) | ∫ ℝ N ∖ B ψ ⁢ ( x + y ) | y | N + 2 ⁢ σ k ⁢ d y | = | ∫ ℝ N ∖ B ( - Δ ) m ⁢ φ ⁢ ( x + y ) | y | N + 2 ⁢ σ k ⁢ d y | ≤ C ⁢ ∥ φ ∥ 2 ⁢ m + 2 , t 1 + | x | N + 2 ⁢ t + C ⁢ ∫ ℝ N ∖ B | φ ⁢ ( x + y ) | | y | N + 2 ⁢ σ k + 2 ⁢ m ⁢ d y .

Moreover,

(A.5) ∫ ℝ N ∖ B | φ ⁢ ( x + y ) | | y | N + 2 ⁢ σ k + 2 ⁢ m ⁢ d y ≤ ∥ φ ∥ 2 ⁢ m + 2 , t 1 + | x | N + 2 ⁢ t ⁢ ∫ ℝ N ∖ B 1 + | x | N + 2 ⁢ t ( 1 + | x + y | N + 2 ⁢ t ) ⁢ | y | N + 2 ⁢ s k ⁢ d y .

By (A.1)–(A.5), it suffices to show that

(A.6) ∫ ℝ N ∖ B 1 + | x | N + 2 ⁢ t ( 1 + | x + y | N + 2 ⁢ t ) ⁢ | y | N + 2 ⁢ s k ⁢ d y < C for all ⁢ x ∈ ℝ N .

If |x|<2, then (A.6) follows by taking the maximum over x∈2⁢B. Fix |x|≥2 and let

U := { y ∈ ℝ N ∖ B : | x + y | ≥ | x | 2 } .

If y∈U, then 1+|x|N+2⁢t≤C⁢(1+|x+y|N+2⁢t), and if y∈ℝN∖U, then |y|>|x|2. Thus,

∫ U 1 + | x | N + 2 ⁢ t ( 1 + | x + y | N + 2 ⁢ t ) ⁢ | y | N + 2 ⁢ s k ⁢ d y ≤ C ⁢ ∫ ℝ N ∖ B | y | - N - 2 ⁢ s k ⁢ d y = C s k < C ,

∫ ℝ N ∖ U 1 + | x | N + 2 ⁢ t ( 1 + | x + y | N + 2 ⁢ t ) ⁢ | y | N + 2 ⁢ s k ⁢ d y ≤ C ⁢ 1 + | x | N + 2 ⁢ t | x | N + 2 ⁢ s k ⁢ ∫ ℝ N 1 1 + | x + y | N + 2 ⁢ t ⁢ d y < C .

This implies (A.6) and finishes the proof. ∎

Lemma A.2.

Let σ∈[0,1], m∈N0, (σk)k∈N⊂[0,1], limk→∞σk=:σ∈[0,1], sk:=m+σk>0, s:=m+σ>0, and φ∈Cc∞⁢(RN). Then (-Δ)sk⁢φ→(-Δ)s⁢φ in Lp⁢(RN) as k→∞ for any p≥1.

Proof.

That (-Δ)sk⁢φ→(-Δ)s⁢φ pointwisely in ℝN as k→∞ follows by using the Fourier transform. In fact, the function t↦(-Δ)t⁢φ⁢(x) is analytic in (0,∞) for every x∈ℝN; see [27, Lemma 2]. Let δ∈(0,s) and p≥1. By Lemma A.1, there is C=C⁢(N,s,δ,φ)>0 such that, for k large enough,

| ( - Δ ) s k ⁢ φ - ( - Δ ) s ⁢ φ | p ≤ C ⁢ ( 1 + | x | N + 2 ⁢ ( s - δ ) ) - p ∈ L 1 ⁢ ( ℝ N ) .

The claim now follows by dominated convergence. ∎

A.2 Uniform Bounds

The goal of this subsection is to show two auxiliary results employed in the proof of Theorem 6.3.

Lemma A.3.

Let s>0, w∈Ds⁢(RN), and η∈Cc∞⁢(RN). Then

w ⁢ η ∈ D s ⁢ ( ℝ N ) 𝑎𝑛𝑑 ∥ w ⁢ η ∥ s ≤ | η ^ | 1 2 ⁢ ∥ w ∥ s .

Proof.

Let η∈Cc∞⁢(ℝN). Then, by the convolution theorem,

∥ w ⁢ η ∥ s 2 = ∫ ℝ N | ξ | 2 ⁢ s ⁢ | w ⁢ η ^ | 2 ⁢ d ξ = ∫ ℝ N | ξ | 2 ⁢ s ⁢ | w ^ * η ^ ⁢ ( ξ ) | 2 ⁢ d ξ

= ∫ ℝ N | ξ | 2 ⁢ s ⁢ | ∫ ℝ N w ^ ⁢ ( ξ - y ) ⁢ η ^ ⁢ ( y ) ⁢ d y | 2 ⁢ d ξ = ∫ ℝ N | ξ | 2 ⁢ s ⁢ | ∫ ℝ N w ^ ⁢ ( ξ ) ⁢ e i ⁢ ξ ⋅ y ⁢ η ^ ⁢ ( y ) ⁢ d y | 2 ⁢ d ξ

≤ ∫ ℝ N | ξ | 2 ⁢ s ⁢ | w ^ ⁢ ( ξ ) | 2 ⁢ | ∫ ℝ N | η ^ ⁢ ( y ) | ⁢ d y | 2 ⁢ d ξ = | η ^ | 1 2 ⁢ ∥ w ∥ s 2 ,

as desired. ∎

Lemma A.4.

Suppose m∈N0, (σk)k∈N⊂[0,1], and limk→∞σk=:σ∈[0,1]. For k∈N, let sk:=m+σk>0, s:=m+σ>0, and wk∈Dsk⁢(RN). If

(A.7) ∥ w k ∥ s k < C 1 for all ⁢ k ∈ ℕ ⁢ and for some ⁢ C 1 > 0 ,

then

∥ ζ n ⁢ w k - w k ∥ s k ≤ C for ⁢ ζ n

as in (6.7) and some C>0 uniform with respect to n and k.

Lemma A.4 follows essentially from Lemma A.5 below. We argue as in [14, Proposition B.1], which shows that ζn⁢w→w as n→∞ in Ds⁢(ℝN) for s∈(0,1). When considering wk∈Dsk⁢(ℝN) instead of w, showing convergence is much more delicate and requires additional assumptions. Here, we only show a uniform bound, which suffices for our purposes.

Let Br⁢(0)=Br denote the ball in ℝN of radius r>0 centered at zero. Let ζn be as in (6.7) and let φn:=1-ζn. Clearly,

(A.8) φ n = 0 ⁢ in ⁢ B n , φ n = 1 ⁢ in ⁢ ℝ N ∖ B 2 ⁢ n , sup x , y ∈ ℝ N ⁡ | ∂ β ⁡ φ n ⁢ ( x ) - ∂ β ⁡ φ n ⁢ ( y ) | | x - y | ≤ C n | β | + 1

for some constant C>0 depending on ζ (given in (6.6)) and a multi-index β∈ℕ0N.

Lemma A.5.

Let m∈N0, σk⊂[0,1], limk→∞σk=:σ∈[0,1], and consider multi-indices α,β∈N0N such that |α|+|β|=m. For k∈N, let sk:=m+σk>0, s:=m+σ>0, and wk∈Dsk⁢(RN) such that (A.7) holds. Then there exists C>0 depending at most on ζ and N such that

∥ ∂ α ⁡ w k ⁢ ∂ β ⁡ φ n ∥ σ k ≤ C for all ⁢ n , k ∈ ℕ .

Proof.

The local case sk∈ℕ is clear, so we may assume that σk∈(0,1) for all k∈ℕ. We begin by splitting into suitable subdomains, more precisely,

∥ ∂ α ⁡ w k ⁢ ∂ β ⁡ φ n ∥ σ = c N , σ k 2 ⁢ ∫ ℝ N ∫ ℝ N | ∂ α ⁡ w k ⁢ ( x ) ⁢ ∂ β ⁡ φ n ⁢ ( x ) - ∂ α ⁡ w k ⁢ ( y ) ⁢ ∂ β ⁡ φ n ⁢ ( y ) | 2 | x - y | N + 2 ⁢ σ k ⁢ d x ⁢ d y

= c N , σ k ⁢ ∫ B 2 ⁢ n ∖ B n ∫ B n ⋯ + c N , σ k ⁢ ∫ ℝ N ∖ B 2 ⁢ n ∫ B n ⋯ + c N , σ k 2 ⁢ ∫ B 2 ⁢ n ∖ B n ∫ B 2 ⁢ n ∖ B n ⋯

+ c N , σ k ∫ ℝ N ∖ B 2 ⁢ n ∫ B 2 ⁢ n ∖ B n ⋯ + c N , σ k 2 ∫ ℝ N ∖ B 2 ⁢ n ∫ ℝ N ∖ B 2 ⁢ n ⋯ + c N , σ k 2 ∫ B n ∫ B n ⋯ = : ∑ i = 1 6 J i .

We show that each Ji is uniformly bounded in k and n for 1≤i≤6. In the following, C>0 denotes possibly different constants depending at most on N, m, σ, and ζ (given in (6.6)).

Estimate for J1. By (A.8),

J 1 = c N , σ k ⁢ ∫ B 2 ⁢ n ∖ B n ∫ B n | ∂ α ⁡ w k ⁢ ( y ) ⁢ ∂ β ⁡ φ n ⁢ ( y ) | 2 | x - y | N + 2 ⁢ σ k ⁢ d x ⁢ d y

= c N , σ k ⁢ ∫ B 2 ⁢ n ∖ B n ∫ B n | ∂ β ⁡ φ n ⁢ ( x ) - ∂ β ⁡ φ n ⁢ ( y ) | 2 | x - y | N + 2 ⁢ σ k ⁢ | ∂ α ⁡ w k ⁢ ( y ) | 2 ⁢ d x ⁢ d y

≤ C ⁢ c N , σ k n 2 ⁢ ( | β | + 1 ) ⁢ ∫ B 2 ⁢ n ∖ B n ∫ B n | ∂ α ⁡ w k ⁢ ( y ) | 2 | x - y | N - 2 ⁢ ( 1 - σ k ) ⁢ d x ⁢ d y .

Note that Bn⊂B3⁢n⁢(y) for y∈B2⁢n∖Bn, and hence

∫ B n 1 | x - y | N - 2 ⁢ ( 1 - σ k ) ⁢ d x ≤ ∫ B 3 ⁢ n ⁢ ( y ) 1 | x - y | N - 2 ⁢ ( 1 - σ k ) ⁢ d x ≤ C 1 - σ k ⁢ n 2 ⁢ ( 1 - σ k ) .

Thus,

J 1 ≤ c N , σ k 1 - σ k ⁢ C n 2 ⁢ ( | β | + σ k ) ⁢ ∫ B 2 ⁢ n ∖ B n | ∂ α ⁡ w k ⁢ ( y ) | 2 ⁢ d y .

By the Fourier transform and the assumption (A.7),

(A.9) ∥ ∂ α ⁡ w k ∥ s k - | α | 2 = ∫ ℝ N | ξ | 2 ⁢ ( s k - | α | ) ⁢ | ∂ α ⁡ w k ^ | 2 ⁢ d ξ = ∫ ℝ N | ξ | 2 ⁢ s k ⁢ | w ^ k | 2 ⁢ d ξ = ∥ w k ∥ s k 2 < C for all ⁢ k ∈ ℕ ,

Then, by Theorem 2.1,

| ∂ α ⁡ w k ⁢ ( y ) | 2 s k - | α | ⋆ < C for all ⁢ k ∈ ℕ

Using Hölder’s inequality, (2.1), and (A.9), we obtain

J 1 ≤ C n 2 ⁢ ( | β | + σ k ) ⁢ ( ∫ B 2 ⁢ n ∖ B n | ∂ α ⁡ w k ⁢ ( y ) | 2 s k - | α | ⋆ ) 2 2 s k - α ⋆ ⁢ ( C ⁢ n N ) 2 ⁢ ( s k - | α | ) N ≤ C ⁢ | ∂ α ⁡ w k ⁢ ( y ) | 2 s k - | α | ⋆ 2 < C

for all n,k∈ℕ, where we have used that |α|+|β|=m and sk=m+σk.

Estimate for J2. If |β|>0, then J2≡0 by (A.8). Thus, let |β|=0. Then |α|=m and, by (A.8),

J 2 = c N , σ k ⁢ ∫ ℝ N ∖ B 2 ⁢ n ∫ B n | ∂ α ⁡ w k ⁢ ( y ) | 2 ⁢ | x - y | - N - 2 ⁢ σ k ⁢ d x ⁢ d y .

By Fubini’s theorem and Hölder’s inequality,

(A.10) J 2 ≤ ∫ B n ( ∫ ℝ N ∖ B 2 ⁢ n | ∂ α ⁡ w k ⁢ ( y ) | 2 σ k ⋆ ⁢ d y ) 2 2 σ k ⋆ ⁢ ( c N , σ k ⁢ ∫ ℝ N ∖ B 2 ⁢ n | x - y | - ( N + 2 ⁢ σ k ) ⁢ N 2 ⁢ σ k ⁢ d y ) 2 ⁢ σ k N ⁢ d x .

Since Bn⁢(x)⊂B2⁢n for every x∈Bn and sk-|α|=σk, we have

c N , σ k ⁢ ∫ ℝ N ∖ B 2 ⁢ n | x - y | - ( N + 2 ⁢ σ k ) ⁢ N 2 ⁢ σ k ⁢ d y ≤ c N , σ k ⁢ ∫ ℝ N ∖ B n ⁢ ( x ) | x - y | - ( N + 2 ⁢ σ k ) ⁢ N 2 ⁢ σ k ⁢ d y ≤ C ⁢ n - N 2 2 ⁢ σ k .

By putting this estimate into (A.10), we obtain, as before, that, for all n,k∈ℕ,

J 2 ≤ C ⁢ n - N ⁢ ( ∫ ℝ N ∖ B 2 ⁢ n | ∂ α ⁡ w k ⁢ ( y ) | 2 σ k ⋆ ⁢ d y ) 2 2 σ k ⋆ ⁢ ( ∫ B n 1 ⁢ d x ) ≤ C ⁢ ( ∫ ℝ N | ∂ α ⁡ w k ⁢ ( y ) | 2 σ k ⋆ ⁢ d y ) 2 2 σ k ⋆ < C .

Estimate for J3. A straightforward computation yields that

J 3 ≤ c N , σ k ⁢ ∫ B 2 ⁢ n ∖ B n ∫ B 2 ⁢ n ∖ B n | ∂ β ⁡ ( x ) ⁡ φ n ⁢ ( x ) - ∂ β ⁡ φ n ⁢ ( y ) | 2 | x - y | N + 2 ⁢ σ k ⁢ | ∂ α ⁡ w k ⁢ ( x ) | 2 ⁢ d x ⁢ d y

+ c N , σ k ⁢ ∫ B 2 ⁢ n ∖ B n ∫ B 2 ⁢ n ∖ B n | ∂ α ⁡ w k ⁢ ( x ) - ∂ α ⁡ w k ⁢ ( y ) | 2 | x - y | N + 2 ⁢ σ k ⁢ | ∂ β ⁡ φ n ⁢ ( y ) | 2 ⁢ d x ⁢ d y .

By using the mean value theorem in the first integral and (A.8) in the second one, we have that

J 3 ≤ c N , σ k n 2 ⁢ ( | β | + 1 ) ⁢ ∫ B 2 ⁢ n ∖ B n ∫ B 2 ⁢ n ∖ B n 1 | x - y | N - 2 ⁢ ( 1 - σ k ) ⁢ | ∂ α ⁡ w k ⁢ ( x ) | 2 ⁢ d x ⁢ d y

+ c N , σ k n 2 ⁢ | β | ∫ B 2 ⁢ n ∖ B n ∫ B 2 ⁢ n ∖ B n | ∂ α ⁡ w k ⁢ ( x ) - ∂ α ⁡ w k ⁢ ( y ) | 2 | x - y | N + 2 ⁢ σ k d x d y = : J 3 ( 1 ) + J 3 ( 2 ) .

An argument similar to the estimate of J1 yields that J3(1)<C uniformly in k and n, whereas

J 3 ( 2 ) ≤ C ⁢ ∥ ∂ α ⁡ w k ∥ σ k < C

for all k,n∈ℕ, by (A.9).

Estimate for J4. If |β|=0, then |α|=m and, by (A.8),

J 4 ≤ c N , σ k ⁢ ∫ ℝ N ∖ B 2 ⁢ n ∫ B 2 ⁢ n ∖ B n | ∂ α ⁡ w k ⁢ ( x ) - ∂ α ⁡ w k ⁢ ( y ) | 2 | x - y | N + 2 ⁢ σ k ⁢ d x ⁢ d y

+ c N , σ k ∫ ℝ N ∖ B 2 ⁢ n ∫ B 2 ⁢ n ∖ B n | φ n ⁢ ( x ) - φ n ⁢ ( y ) | 2 | x - y | N + 2 ⁢ σ k | ∂ α w k ( x ) | 2 d x d y = : J 4 ( 1 ) + J 4 ( 2 ) .

Note that J4(1)≤C⁢∥∂α⁡wk∥σk<C for all k,n∈ℕ as in the previous step. On the other hand, by the mean value theorem,

J 4 ( 2 ) ≤ C ⁢ c N , σ k ⁢ ∫ ℝ N ∖ B 3 ⁢ n ∫ B 2 ⁢ n ∖ B n | ∂ α ⁡ w k ⁢ ( x ) | 2 | x - y | N + 2 ⁢ σ k ⁢ d x ⁢ d y + C ⁢ c N , σ k n 2 ⁢ ∫ B 3 ⁢ n ∖ B 2 ⁢ n ∫ B 2 ⁢ n ∖ B n | ∂ α ⁡ w k ⁢ ( x ) | 2 | x - y | N - 2 ⁢ ( 1 - σ k ) ⁢ d x ⁢ d y

and the uniform bounds follow as in the estimates for J2 and J1, respectively.

If |β|>0, we see that

J 4 = c N , σ k ⁢ ∫ ℝ N ∖ B 2 ⁢ n ∫ B 2 ⁢ n ∖ B n | ∂ β ⁡ φ n ⁢ ( x ) - ∂ β ⁡ φ n ⁢ ( y ) | 2 | x - y | N + 2 ⁢ σ k ⁢ | ∂ α ⁡ w k ⁢ ( x ) | 2 ⁢ d x ⁢ d y ,

and the uniform bound follows as in the estimate for J2.

Estimate for J5. If |β|>0, then J5=0. If |β|>0, then, by (A.9), for all n,k∈ℕ,

J 5 = c N , σ k 2 ⁢ ∫ ℝ N ∖ B 2 ⁢ n ∫ ℝ N ∖ B 2 ⁢ n | ∂ α ⁡ w k ⁢ ( x ) - ∂ α ⁡ w k ⁢ ( y ) | 2 | x - y | N + 2 ⁢ σ k ⁢ d x ⁢ d y ≤ ∥ ∂ α ⁡ w k ∥ σ k < C .

The proof is then finished since J6≡0 by (A.8). ∎

Proof of Lemma A.4.

We show the case m∈ℕ even and (σk)k∈ℕ∈(0,1). The other cases follow similarly. Since m is even,

( - Δ ) m 2 ⁢ ( u ⁢ v ) = ( - Δ ) m 2 ⁢ u ⁢ v + ∑ { α , β : | α | + | β | = m , | α | < m } μ α , β ⁢ ∂ α ⁡ u ⁢ ∂ β ⁡ v for ⁢ u , v ∈ H m ⁢ ( ℝ N )

and for suitable coefficients μα,β∈ℕ0. Then, for sk=m+σk and wk∈Dsk⁢(ℝN),

(A.11) ∥ w k ⁢ φ n ∥ s k = ∥ ( - Δ ) m 2 ⁢ ( w k ⁢ φ n ) ∥ σ k ≤ ∥ ( - Δ ) m 2 ⁢ w k ⁢ φ n ∥ σ k + C ⁢ ∑ { α , β : | α | + | β | = m , | α | < m } ∥ ∂ α ⁡ w k ⁢ ∂ β ⁡ φ n ∥ σ k

for some C=C⁢(m,N)>0, where φn is given in (A.8). The claim now follows from (A.11) and Lemma A.5. ∎

Acknowledgements

We also thank Prof. Mónica Clapp (IMUNAM) for fruitful discussions.

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Received: 2021-02-17

Revised: 2021-05-19

Accepted: 2021-05-19

Published Online: 2021-07-28

Published in Print: 2021-11-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Existence and Convergence of Solutions to Fractional Pure Critical Exponent Problems

Abstract

1 Introduction

Theorem 1.1.

Proposition 1.2.

Theorem 1.3.

2 Preliminaries

2.1 Functional Framework

Theorem 2.1 (Fractional Sobolev Inequality).

Proof.

Theorem 2.2.

Proof.

2.2 Symmetric Setting

Lemma 2.3.

Proof.

Lemma 2.4.

Proof.

Lemma 2.5.

Proof.

Lemma 2.6.

2.3 Groups and Homomorphism for Sign-Changing Solutions

3 Uniform Bounds and Asymptotic Estimates

Lemma 3.1.

Proof.

Lemma 3.2.

Proof.

Lemma 3.3.

Proof.

Lemma 3.4.

Proof.

Lemma 3.5.

Proof.

Lemma 3.6.

Proof.

4 A Concentration Result

Theorem 4.1.

Proof.

5 Existence, Nonexistence, and Convergence of Solutions in Symmetric Bounded Domains

Proof of Proposition 1.2.

Proposition 5.1.

Proof.

Lemma 5.2.

Proof.

Theorem 5.3.

Proof.

Proof of Theorem 1.1.

6 Existence and Convergence of Entire Solutions

Theorem 6.1.

Proof.

Lemma 6.2.

Proof.

Theorem 6.3.

Proof.

Proof of Theorem 1.3.

A Auxiliary Lemmas

A.1 Convergence of Test Functions

Lemma A.1.

Proof.

Lemma A.2.

Proof.

A.2 Uniform Bounds

Lemma A.3.

Proof.

Lemma A.4.

Lemma A.5.

Proof.

Proof of Lemma A.4.

Acknowledgements

References

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