Existence and concentration of positive solutions for a critical p&q equation

Gustavo S. Costa; Giovany M. Figueiredo

doi:10.1515/anona-2020-0190

Open Access Published by De Gruyter July 17, 2021

Existence and concentration of positive solutions for a critical p&q equation

Gustavo S. Costa and Giovany M. Figueiredo

From the journal Advances in Nonlinear Analysis

https://doi.org/10.1515/anona-2020-0190

Abstract

We show existence and concentration results for a class of p&q critical problems given by

−divaϵp|∇u|pϵp|∇u|p−2∇u+V(z)b|u|p|u|p−2u=f(u)+|u|q⋆−2uinRN,

where u ∈ W^1,p(ℝ^N) ∩ W^1,q(ℝ^N), ϵ > 0 is a small parameter, 1 < p ≤ q < N, N ≥ 2 and q^* = Nq/(N − q). The potential V is positive and f is a superlinear function of C¹ class. We use Mountain Pass Theorem and the penalization arguments introduced by Del Pino & Felmer’s associated to Lions’ Concentration and Compactness Principle in order to overcome the lack of compactness.

Keywords: Critical exponent; p&q Laplacian operator; Variational methods

MSC 2010: Primary 35J60; Secondary 35J60; 35J10; 35J20

1 Introduction

In this paper we are concerned with a class of problems, named p&q problems type. In the last years the main interest in this general class of problems has been due to the fact that they arise from applications in physics and related sciences, such as biophysics, plasma physics and chemical reaction, as it can be seen for example in [20], [23] and [35]. In addition, such a class of problems encompasses a large class of problems, as can be seen in [4], [15] and [17].

More precisely, we show existence and concentration results of positive solutions for the critical problem given by

(Pε) −divaϵp|∇u|pϵp|∇u|p−2∇u+V(z)b|u|p|u|p−2u=f(u)+|u|q∗−2u in RN,u∈W1,pRN∩W1,qRN,

where ϵ > 0, N ≥ 2, 1 < p ≤ q < N and q^* = Nq/(N − q). The hypotheses on the function a are the following: (a₁)the function a is of class C¹ and there exist constants k₁, k₂ ≥ 0 such that

k1tp+tq≤atptp≤k2tp+tq, for all t>0;

(a₂)the mapping t↦a(tp)tq−p is nonincreasing for t > 0;

(a₃)if 1 < p < 2 ≤ N the mapping t ↦ a(t) is nondecreasing for t > 0. If 2 ≤ p < N the mapping t↦atptp−2 is nondecreasing for t > 0.

As a direct consequence of (a₂) we obtain that the map a and its derivative a^′ satisfy

(1.1) a′(t)t≤(q−p)pa(t)for allt>0.

Now if we define the function h(t)=a(t)t−qpA(t), using (1.1) we can prove that the function h is nonincreasing. Then, there exists a positive real constant γ≥qp such that

(1.2) 1γa(t)t≤A(t),for allt≥0.

The hypotheses on the function b are the following:

(b₁)The function b is of class C¹ and there exist constants k₃, k₄ ≥ 0 such that

k3tp+tq≤b(tp)tp≤k4tp+tq,for allt>0;

(b₂)the mapping t↦b(tp)tq−p is nonincreasing for t > 0.

(b₃)if 1 < p < 2 ≤ N the mapping t ↦ b(t) is nondecreasing for t > 0. If 2 ≤ p < N the mapping t↦btptp−2 is nondecreasing for t > 0.

Using the hypothesis (b₂) and arguing as (1.1) and (1.2) we prove that there exists γ≥qp such that

(1.3) 1γb(t)t≤B(t),for allt≥0.

The nonlinearity f is assumed to be a C¹ function with the following hypotheses:

(f₁)

lim|s|→0f(s)|s|q−1=0.

(f₂) There exists q < r < q ∗ = q N N − q such that

lim|s|→∞f(s)|s|r−1=0.

(f₃) There exists θ∈(γp,q∗) such that

0<θF(s)≤f(s)sfor s>0,

where F(s)=∫0sf(t)dt and γ > 0 was given in (1.2);

(f₄) s↦f(s)sq−1 is nondecreasing for s > 0.

(f₅) There exist τ ∈ (q, q*) and λ > 1

f(s)≥λsτ−1 ∀ s>0.

We need to put some hypotheses on the potential V ∈ C(ℝ^N).

(V₁)There is V₀ > 0, such that

0 < V 0 ≤ V ( z ) for all z ∈ R N .

(V₂)There exists a bounded domain Ω ⊂ ℝ^N such that

0 < V 0 = inf z ∈ Ω V ( z ) < inf z ∈ ∂ Ω V ( z ) .

In order to illustrate the degree of generality of the kind of problems studied here, with adequate hypotheses on the functions a and b, in the following we present more some examples of problems which are also interesting from the mathematical point of view and have a wide range of applications in physics and related sciences.

Problem 1

Let a(t)=1+tq−pp and b(t)=1+tq−pp. In this case we are studying problem

−Δpu−Δqu+V(x)(|u|p−2u+|u|q−2u)=f(u)+|u|q∗−2uin RN.

The Problem 1 comes from a general reaction–diffusion system: u_t = div(Du∇u) + g(x, u), where Du:=[|∇u|p−2+|∇u|q−2]. In such applications, the function u describes a concentration, the term div(Du∇u) corresponds to the diffusion with a diffusion coefficient Du and g(·, u) is the reaction and relates to source and loss processes. Usually, in chemical and biological applications, the reaction term g(·, u) is a polynomial of u with variable coefficients.

Problem 2

Let a(t)=tq−pp and b(t)=tq−pp. In this case we are studying problem

−ϵqΔqu+V(x)|u|q−2u=f(u)+|u|q∗−2uinRN

and it is related to the main result showed in [3] in the case q = 2. In [19] the author have studied the case 1 < q < N.

Problem 3

Let a(t)=1+1(1+t)p−2pandb(t)=1. In this case we are studying problem

− ϵ p div ⁡ ( | ∇ u | p − 2 ∇ u ) − d i v ϵ p | ∇ u | p − 2 ∇ u ( 1 + ϵ p | ∇ u | p ) p − 2 p + V ( x ) | u | p − 2 u = f ( u ) + | u | q ∗ − 2 u in R N .

Problem 4

Let a(t)=1+tq−pp+1(1+t)p−2p and b(t)=1+tq−pp. In this case we are studying problem

− ϵ p Δ p u − ϵ q Δ q u − div ϵ p | ∇ u | p − 2 ∇ u ( 1 + ϵ p | ∇ u | p ) p − 2 p + V ( x ) ( | u | p − 2 u + | u | q − 2 u ) = f ( u ) + | u | q ∗ − 2 u in R N .

The main result is the following:

Theorem 1.1

Suppose that a, b, f and V satisfy (a₁)− (a₃), (b₁)− (b₃), (f₁)− (f₅) and (V₁)− (V₂) respectively. Then there are ϵ₀ > 0 and λ^* > 1 such that (P_ϵ) has a positive solution w_ϵ ∈ W^1,p(ℝ^N) ∩ W^1,q(ℝ^N), for every ϵ ∈ (0, ϵ₀) and for every λ > λ^*. In addition, if P_ϵ is the maximum point of w_ϵ, then

limϵ→0V(Pϵ)=V0.

Moreover, there are positive constants C and α such that

|wϵ(z)|≤Cexp(−α|z−Pϵϵ|),

for all ϵ ∈ (0, ϵ₀) and for all z ∈ ℝ^N.

In a seminal paper [31], Rabinowitz used his famous Mountain Pass Theorem(joint with Ambrosetti) [5] and showed the existence of solution for a Nonlinear Schrödinger Equation given by

(R) −ϵ2Δu+V(x)u=f(u) inRN, u>0inRN,

where V is a continuous potential satisfying (V₁) and

[(R1)]lim inf|x|→∞V(x)=V∞, where V_∞ < ∞or V_∞ = ∞.

In [31], Rabinowitz used the force of the parameter ϵ and the geometry of the potential V in order to overcome the lack of compactness of Sobolev’s embedding to obtain the positive solution. In [33], Wang showed that the solution found by Rabinowitz concentrates around a local minimum of the potential V,when ϵ converges to zero. Wang also noted that the concentration of any family of solutions with energy uniformly bounded can only occur in a critical point of V. In [12], Del Pino and Felmer weakened the hypothesis (R₁) of Rabinowitz and created a method that is known as Del Pino and Felmer’s penalization method.

As can be seen in [4], [15] and [17], p&q problems are generalizations of (R). However, as can seen below, we show that the arguments found in [12], [31] and [33] cannot be used directly. But before that, we are going to report some results on p&q problems type. There are interesting papers on such class of problems. We start with some problems in a bounded domain. For example, in [15] the author shows the existence and multiplicity of solutions for a critical p&q problem considering nonlinearity of type concave and convex. The critical case with discontinuous nonlinearities has studied in [16].

Now we comment some results in ℝ^N. Existence results was studied in [11] and [17]. In [2] the authors studied concentration results in Orlicz-Sobolev spaces with subcritical nonlinearity and the potential satisfying the local condition introduced by Del pino and Felmer [12]. In [4], it was showed the existence and concentration results with subcritical nonlinearity and the potential satisfying the global condition introduced by Rabinowitz [31]( see also [33]).

The present work is strongly influenced by the articles above. Below we list what we believe that are the main contributions of our paper.

Unlike [4], [11] and [17], we show existence and concentration results considering the local hypothesis on potential introduced by Del Pino and Felmer [12].
Unlike [2], we are considering the critical nonlinearity.
Since the operator is not homogeneous, some estimates are different and more delicate than some estimates that can be found in [12] and [31] . For example, see Lemma 3.4, Proposition 5.1, Lemma 5.7 and all the Lemmas of Section 7.
In order to overcome the lack of compactness provoked by the critical growth, it is very common to use the Talenti’s function (see [32]) to have some control on the minimax level, as can be seen in [10, Lemma 1.1]. The lack of homogenity of the p&q operator does not allow to use this argument. We overcome this difficulty using the solution of a problem in a bounded domain, as can be seen in Lemma 3.5.

The interest in the study of nonlinear partial differential equations with p&q operator or fractional p&q operator has increased because many applications arising in mathematical physics may be stated with an operator in this form. We cite the papers [6], [7], [8], [9], [18], [21], [22], [26], [27], [28], [29], [30] and their references. Several techniques have been developed or applied in their study, such as variational methods, fixed point theory, lower and upper solutions, global branching, and the theory of multivalued mappings.

This paper is organized as follows. In Section 2, we define an auxiliary problem using the penalization argument introduced by Del Pino and Felmer [12]. The existence of solution for the auxiliary problem was showed in Section 3. In order to show the concentration result, in Section 4 we studied the autonomous problem. The concentration result was showed in Section 5. In Section 6 we showed that the solutions of the auxiliary problem are solutions of the original problem. In Section 7 we showed the exponential decay of these solutions. To conclude the paper, we showed in an appendix the existence of a solution to a problem in a bounded domain that was important to overcome the lack of compactness.

2 Variational framework and an auxiliary problem

To prove Theorem 1.1, we will work with the problem below, which is equivalent to (P_ϵ) by change variable z = ϵx, which is given by

(P~ε) −divϵa|∇u|p|∇u|p−2∇u+V(ϵx)b|u|p|u|p−2u=f(u)+|u|q∗−2u in RN,u∈W1,pRN∩W1,qRN,

where ϵ > 0, N ≥ 2 and 1 < p ≤ q < N.

In order to obtain solutions of (P˜ϵ), consider the following subspace of W1,p(RN)⋂W1,q(RN) given by

W ϵ := { v ∈ W 1 , p ( R N ) ∩ W 1 , q ( R N ) : ∫ R N V ( ϵ x ) b ( | v | p ) | v | p d x < + ∞ } ,

which is a Banach space when endowed with the norm

∥u∥=∥u∥1,p+∥u∥1,q,

where

∥ u ∥ 1 , m = ∫ R N | ∇ u | m d x + ∫ R N V ( ϵ x ) | u | m d x 1 m , for m ≥ 1.

Since the approach is variational, consider the energy functional associated J_ϵ : W_ϵ → ℝ given by

Jϵ(v)=1p∫RNA|∇v|pdx+1p∫RNV(ϵx)B|v|pdx−∫RNF(v)dx−1q∗∫RNv+q∗dx,

where u₊ = max{u, 0}. By standard arguments, one can prove that J_ϵ ∈ C¹(W_ϵ , ℝ). As we are interested in nonnegative solutions we can assume that f (s) = 0 for s ≤ 0.

Let β be a positive number satisfying β>maxpγθq(θ−pγ),V0pγq,1, where θ was given in (f₃) and V₀ appeared in (V₁). From (f₄), there exists η > 0 such that f(η)+ηq∗−1ηq−1=V0β. Then, using the above numbers,

we define the function of C¹ class given by

f ~ ( s ) = 0 i f s ≤ 0 , f ( s ) + s q ∗ − 1 i f 0 < s ≤ η 2 , V 0 β | s | q − 2 s i f s > η .

We now define the function

g(z,s):=χΩ(z)[f(s)+(s+)q∗−1]+(1−χΩ(z))f˜(s),

and the auxiliary problem

(Pε_aux) −divϵa|∇u|p|∇u|p−2∇u+V(ϵx)b|u|p|u|p−2u=g(ϵx,u) in RN,u∈Wϵ

where χ_Ω is the characteristic function of the set Ω. It is easy to check that (f₁)− (f₄) imply that g is a Carathéodory function and for x ∈ ℝ^N, the function s → g(ϵx, s) is of class C¹ and satisfies the following conditions, uniformly for x ∈ ℝ^N:

(g₁) lim|s|→0g(ϵx,s)|s|q−1=0

(g₂) g(ϵx,s)≤f(s)+sq∗−1, ∀ s>0 and x∈RN

(g₃)_i 0<θG(ϵx,s)≤g(ϵx,s)s,∀ϵx∈Ωand∀s>0

and

(g₃)_ii 0 < q G ( ϵ X , s ) ≤ g ( ϵ X , s ) s ≤ 1 β V ( ϵ x ) | s | q , ∀ ϵ X ∉ Ω and ∀ s > 0 ,

where G(ϵx,s)=∫0sg(ϵx,t)dt.

The function

(g₄) s→g(ϵx,s)|s|q−1is nondecreasing.

Remark 1

Note that, for z = ϵx, if u_ϵ is a positive solution of (P_ϵaux ) with |uϵ(z)|≤η2 for every ϵx ∈ ℝ^N \ Ω, then u_ϵ(x) is also a positive solution of (P_ϵ).

3 Existence of ground state for problem (P_ϵaux)

Hereafter, let us denote by I_ϵ : W_ϵ → ℝ the functional given by

Iϵ(v)=1p∫RNA|∇v|pdx+1p∫RNV(ϵx)B|v|pdx−∫RNG(ϵx,v)dx.

We denote by 𝒩_ε the Nehari manifold of I_ε, that is,

N ε := { u ∈ W ε ∖ { 0 } : 〈 I ε ′ ( u ) , u 〉 = 0 }

and define the number b_ε by setting

(3.1) bε:=infu∈NεIε(u).

Using (f₁), (f₂) and (g₂) we have: for every ξ > 0 there exists C _ξ such that

(3.2) |g(εx,s)|≤ξ|s|q−1+Cξ|s|r−1+|s|q∗−1 for all x∈RN, s∈R.

Then, by definition of g and (3.2), there is r_ε > 0 such that

(3.3) ∥u∥≥rε>0 for all u∈Nε.

The main result in this section is:

Theorem 3.1

Let a satisfying (a₁)−(a₃), b satisfying (b₁)−(b₃), f satisfying (f₁)−(f₅) and V such that (V₁)−(V₂) hold. Then, there is λ^* > 1 such that (P_ϵaux ) has positive solution uϵ∈W1,p(RN)∩W1,q(RN),foreveryλ>λ∗.

Moreover, we would like to highlight that in section 5, more precisely in Lemma 5.5, we are going to show that if Pϵϵ is the maximum point of u_ϵ then

limϵ→0V(Pϵ)=V0.

In order to use the Mountain Pass Theorem [5], we define the Palais-Smale compactness condition. We say that a sequence (u_n) ⊂ W_ϵ is a Palais-Smale sequence at level c for the functional I_ϵ if

Iϵ(un)→cand∥Iϵ′(un)∥→0in (Wϵ)′,

where

c:=infη∈Γmaxt∈[0,1]Iϵ(η(t))>0andΓ:={η∈C([0,1],X):η(0)=0, Iϵ(η(1))<0}.

If every Palais-Smale sequence of I_ϵ has a strong convergent subsequence, then one says that I_ϵ satisfies the Palais-Smale condition ((PS) for short).

Lemma 3.2

The functional I_ϵ : W_ϵ → ℝ satisfies the following conditions

There are α, ρ > 0 such that

Iϵ(u)≥α, if ∥u∥=ρ.

For any u∈C0∞(Ωϵ,[0,∞)), we have

limt→∞Iϵ(tu)=−∞.

Proof. Using (a₁), (b₁) and (3.2) we obtain

I ϵ ( u ) ≥ min { k 1 , k 3 } p ∥ u ∥ 1 , p p + 1 q ∥ u ∥ 1 , q q − ξ p ∫ R N | u | p d x − C ξ r ∫ R N | u | r d x − 1 q ∗ ∫ R N | u | q ∗ d x .

By Sobolev embeddings, choosing ξ > 0 appropriate and taking ‖u‖ < 1 there are positive constants C₁, C₂, C₃, such that

Iϵ(u)≥C1∥u∥1,pp+∥u∥1,qq−C2∥u∥r−C3∥u∥q∗≥C4∥u∥q−C2∥u∥r−C3∥u∥q∗.

Then the item (i) follows.

Now we show that the item (ii) holds. Consider a positive function w∈C0∞(Ωϵ),t>0 and using (a₁), (b₁), (f₃) and Sobolev embedding, we have

Iϵ(tw)≤tppmax{k2,k4}∥w∥1,pp+tqq∥w∥1,qq−tq∗q∗∫Ωϵ|w|q∗dx.

This proves the second item.

Hence, there exists a Palais-Smale sequence (u_n) ⊂ W_ϵ at level c_ϵ. Using (a₂), (b₂) and (f₄), it is possible to prove that

c ϵ = b ϵ = inf u ∈ W ϵ ∖ { 0 } sup t ≥ 0 I ϵ ( t u ) ,

where b_ϵ was defined in (3.1).

In order to prove the Palais-Smale condition, we need to prove the next lemma.

Lemma 3.3

Let (u_n) be a (PS)_d sequence for I_ϵ, then the sequence (u_n) is bounded W_ϵ. Moreover, for each ξ > 0 there exists R = R(ξ) > 0 such that

lim sup n → ∞ ∫ R N ∖ B R ( 0 ) [ a ( | ∇ u n | p ) | ∇ u n | p + V ( ϵ x ) b ( | u n | p ) | u n | p ] d x < ξ .

Proof. Since (u_n) is a (PS)_d sequence for functional I_d, then using (1.1), (1.3), (g₃)_i and (g₃)_ii we have that

on(1)+d+on(1)∥un∥=Iϵ(un)−1θIϵ′(un)un≥1pγ−1θ∫RNa(|∇un|p)|∇un|p+1+μV(x)b(|un|p)|un|pdx−1β∫RN|∇u|q+V(ϵx)|u|qdx≥1pγ−1θmin{k1,k3}∥un∥1,pp+1−1β∥un∥1,qq.

Then, arguing as the [4, Lemma 2.3] , we can concluded that (u_n) is bounded in W_ϵ.

Let η_R ∈ C^∞(ℝ^N) be such that η_R(x) = 0 if x ∈ B_R_/2(0) and η_R(x) = 1 if x ∉ B_R(0), with 0 ≤ η_R(x) ≤ 1 and |∇ηR|≤CR, where C is a constant independent of R. Since the sequence (η_Ru_n) is bounded in W_ϵ, and fixing R > 0 such that Ω_ϵ ⊂ B _R_/2(0) we obtain, by definition of the functional I_ϵ,

∫ R N ∖ B R ( 0 ) [ a ( | ∇ u n | p ) | ∇ u n | p + V ( ϵ x ) b ( | u n | p ) | u n | p ] d x = I ϵ ( u n ) u n η R + ∫ R N g ( ϵ x , u n ) u n η R d x − ∫ R N u n a ( | ∇ u n | p ) | ∇ u n | p − 2 ∇ u n ∇ η R d x + o n ( 1 ) .

Using (g₃)_ii we estimate

1 − 1 β ∫ R N ∖ B R ( 0 ) [ a ( | ∇ u n | p ) | ∇ u n | p + V ( ϵ x ) b ( | u n | p ) | u n | p ] d x ≤ ∫ R N | u n | a ( | ∇ u n | p ) | ∇ u n | p − 1 | ∇ η R | d x + o n ( 1 ) .

As (u_n) is bounded in W_ϵ and |∇ηR|≤CR. Passing to the limit in the last estimate, we get

lim sup n → ∞ ∫ R N ∖ B R [ a ( | ∇ u n | p ) | ∇ u n | p + V ( ϵ x ) b ( | u n | p ) | u n | p ] d x < ξ .

for some R sufficiently large and for some fixed ξ > 0.

In the next result we show that the functional I_ϵ satisfies the Palais-Smale condition for some levels. For this work we are denoting by S the best Sobolev constant for the embedding of D^1,q(ℝ^N) into Lq∗(RN), that is, the largest positive constant S such that

(3.4) S ∫ R N | u | q ∗ d x q q ∗ ≤ ∫ R N | ∇ u | q d x for every u ∈ D 1 , q ( R N ) .

Lemma 3.4

The functional I_ϵ satisfies the Palais-Smale condition at any level

d<1θ−1q∗SN/q.

Proof. Let (u_n) ⊂ W_ϵ be a Palais-Smale sequence at level d<1θ−1q∗SN/q for the functional I_ϵ. Arguing as Lemma [4, Lemma 2.3] we have that (u_n) is bounded in W_ϵ. Then by Sobolev embeddings we deduce, up to a subsequence, that

(3.5) un⇀u weakly in Wϵ,∇un(x)→∇u(x) q.t.p in RN,un→u strongly in Llocs(RN) for any p≤s<q∗,un(x)→u(x) for a.e x∈RN.

Using the same kind of ideias contained [4, Lemma 2.3], we may conclude that u is a critical point of I_ϵ. From Lemma 3.3 and for each ξ > 0 given there exists R > 0 such that

lim sup n → ∞ ∫ R N ∖ B R ( 0 ) [ a ( | ∇ u n | p ) | ∇ u n | p + V ( ϵ x ) b ( | u n | p ) | u n | p ] d x < ξ .

This inequality, (a₁), (b₁), (f₁), (f₂), (g₂) and the Sobolev embeddings imply, for n large enough, there exists a positive constant C₁ such that

(3.6) ∫ R N ∖ B R ( 0 ) g ( ϵ x , u n ) u n d x ≤ C 1 ξ + ξ r / q + ξ q ∗ / q .

On the other hand, taking R large enough, we suppose that

(3.7) ∫ R N ∖ B R ( 0 ) g ( ϵ x , u ) u d x < ξ .

Therefore, by (3.6) and (3.7),

(3.8) ∫ R N ∖ B R ( 0 ) g ( ϵ x , u n ) u n d x = ∫ R N ∖ B R ( 0 ) g ( ϵ x , u ) u d x + o n ( 1 ) .

We claim that

(3.9) ∫ B R ( 0 ) ∩ ( R N ∖ Ω ϵ ) g ( ϵ x , u n ) u n d x = ∫ B R ( 0 ) ∩ ( R N ∖ Ω ϵ ) g ( ϵ x , u ) u d x + o n ( 1 ) .

Indeed, we have, in view of the definition of g,

g(ϵx,un)un≤f(un)un+η2q∗+V0β|un|q for any x∈RN∖Ωϵ.

Since the set B_R(0) ∩ (ℝ^N\Ω_ϵ) is bounded we can use the above estimate, (f₁), (f₂), (3.5) and Lebesgue’s Theorem to conclude that the convergence (3.9) holds.

Finally, we now prove the following convergence

(3.10) ∫ Ω ϵ | u n | q ∗ d x = ∫ Ω ϵ | u | q ∗ d x + o n ( 1 ) .

Since (u_n) is bounded in W_ϵ and using the Lions’s Concentration Compactness Principle [25],wemay suppose that

|∇un|q⇀μ and |un|q∗⇀ν.

Then we obtain an at most countable index set Γ, sequences (x_i) ⊂ ℝ^N and (μ_i), (ν_i) ⊂ (0,∞), such that

(3.11) μ ≥ | ∇ u | q + ∑ i ∈ Γ μ i δ x i , ν = | u | q ∗ + ∑ i ∈ Γ ν i δ x i and S ν i q / q ∗ ≤ μ i ,

for all i ∈ Γ,where δxi is the Dirac mass at x_i ∈ ℝ^N. Thus it is sufficient to show that {xi}i∈Γ∩Ωϵ=∅. Then, we suppose by contradiction that x_i ∈ Ω_ϵ for some i ∈ Γ. Consider R > 0 and the function ψ_R := ψ(x_i − x), where ψ∈C0∞(RN,[0,1]) is such that ψ ≡ 1 in B_R(x_i), ψ ≡ 0 in ℝ^N\B_2R(x_i), |∇ψ|_∞ ≤ 2, where R > 0 will be chosen in such way that the support of ψ is contained in Ω_ϵ. Then, as (ψ_Ru_n) is bounded and Iϵ′(un)ψRun=on(1),

∫ R N u n a ( | ∇ u n | p ) | ∇ u n | p − 2 ∇ u n ⋅ ∇ ψ R d x + ∫ R N ψ R a ( | ∇ u n | p ) | ∇ u n | p d x + ∫ R N ψ R V ( ϵ x ) b ( | u n | p ) | u n | p d x = ∫ R N f ( x , u n ) ψ R u n d x + ∫ R N ψ R | u n | q ∗ d x + o n ( 1 ) .

Note that, using (a₁), (b₁) and that the function f has subcritical growth, we have

lim R → 0 lim n → ∞ ∫ R N u n a ( | ∇ u n | p ) | ∇ u n | p − 2 ∇ u n p ⋅ ∇ ψ R d x = 0 , lim R → 0 lim n → ∞ ∫ R N V ( ϵ x ) b ( | u n | p ) | u n | p ψ R d x = 0 ,

and

limR→0limn→∞∫RNf(x,un)ψRundx=0.

Therefore, by (a₁) again,

∫ R N ψ R | ∇ u n | q d x ≤ ∫ R N | u n | q ∗ ψ R d x + o n ( 1 ) .

Since ψ_R has compact support and letting n → ∞in the above expression, we see that

∫RNψRdμ≤∫RNψRdν,

which implies

μi≤νi.

From this inequality and (3.11) one easily sees that SN/q≤νi.Asβ>pγθq(θ−pγ)andSN/q≤νi we have, by previous arguments,

d=Iϵ(un)−1θIϵ′(un)un+on(1)≥θ−pγpγθ−1qβ∥un∥1,qq+1θ−1q∗∫Ωϵ|un|q∗dx+on(1)≥1θ−1q∗∫ΩϵψR|un|q∗dx+on(1).

Hence, taking the limit and using (3.11), we get

d≥1θ−1q∗∑i∈ΓψR(xi)νi=1θ−1q∗νi≥1θ−1q∗SN/q

which does not make sense. Thus we obtain the convergence (3.10).

Therefore

(3.12) ∫RNg(ϵx,un)undx=∫RNg(ϵx,u)udx+on(1).

Finally, we prove that, up to a subsequence, u_n → u in W_ϵ. Since Iϵ′(un)un=on(1),Iϵ′(u)=0 (3.12) and Fatou’s Lemma we have

0≤∫RNa(|∇un|p)|∇un|p−a(|∇u|p)|∇u|pdx+∫RNV(ϵx)b(|un|p)|un|p−b(|u|p)|u|pdx+∫RNg(ϵx,u)u−g(ϵx,un)undx=on(1).

Then, using (a₁) and (b₁), we obtain ‖u_n − u‖ = o_n(1), that is, the sequence (u_n) converges strongly to u.

For each fixed ϵ > 0, let us consider the following problem

(Pτ) −k2Δpu−Δqu+Vk4|u|p−2u+|u|q−2u=|u|τ−2u in Ωε,u∈W01,qΩϵ,

where τ is the constant which appears in the hypothesis (f₅) and V:=maxx∈Ωϵ‾V(x) is a positive constant. We have associated to problem (P_τ) the functional

Iτ(u)=1p∫Ωϵk2|∇u|p+Vk4|u|pdx+1q∫Ωϵ|∇u|q+V|u|qdx−1τ∫Ωϵ|u|τdx

and the associated Nehari manifold

Nτ={u∈W01,q(Ωϵ): u≠0 and Iτ′(u)u=0}.

From Appendix there exists wτ∈W01,q(Ωϵ) such that

Iτ(wτ)=cτ:=infu∈NτIτ(u),Iτ′(wτ)=0

and

(3.13) c τ ≥ τ − q τ q ∫ R N | w τ | τ d x .

Since λ is the parameter which appears in the hypothesis (f₅) we have the following result.

Lemma 3.5

There exists λ^* > 1, such that if λ>λ∗,thencϵ<(1θ−1q∗)SN/q.

Proof. First of all, by the hypotheses (a₁), (b₁) and (f₅), we obtain

∫ R N a ( | ∇ w τ | p ) | ∇ w τ | p d x + ∫ R N V ( ϵ x ) b ( | w τ | p ) | w τ | p d x ≤ ∫ Ω ϵ k 2 | ∇ w τ | p + V k 4 | w τ | p d x + ∫ Ω ϵ | ∇ w τ | q + V | w τ | q d x = ∫ Ω ϵ | w τ | τ d x ≤ ∫ Ω ϵ f ( w τ ) w τ d x ≤ ∫ R N g ( ϵ x , w τ ) w τ d x ,

where V:=maxx∈Ωϵ‾V(x) This inequality implies that Iϵ′(wτ±)wτ±≤0, and then there exists t ∈ (0, 1) such that tw_τ ∈ 𝒩_ϵ. Using (a₁), (b₁) and (f₅), we obtain

cϵ≤Iϵ(twτ)≤tpp∫Ωϵk2|∇wτ|p+Vk4|wτ|pdx+tqq∫Ωϵ|∇wτ|q+V|wτ|qdx−λτtτ∫Ωϵ|wτ|τdx.

Since t ∈ (0, 1), p ≤ q and Iϵ′(wτ)wτ=0, we get

cϵ≤Iϵ(twτ)≤tpp∫Ωϵk2|∇wτ|p+Vk4|wτ|pdx+tpp∫Ωϵ|∇wτ|q+V|wτ|qdx−λτtτ∫Ωϵ|wτ|τdx=tpp−λtττ∫Ωϵ|wτ|τdx≤maxs≥0spp−λsττ∫Ωϵ|wτ|τdx.

Using (3.13), we have

cϵ≤maxs≥0spp−λsττcτqτ(τ−q)≤τ−ppλp/(τ−p)cτq(τ−q)

By some straightforward algebric manipulations, we get

cϵ≤τ−ppλp/(τ−p)cτq(τ−q).

Then, if we choose λ>λ∗:=max1,[(τ−p)(τ−q)qpθq∗(q∗−θ)cτSN/q](τ−p)/p in the hypothesis (f₅), the proof is complete.

3.1 Proof of the Theorem 3.1

Proof. The proof is a consequence of Lemma 3.2, Lemma 3.4 and Lemma 3.5.

4 The Autonomous Problem

In order to prove the concentration result, we consider the following problem

(P_o) −diva|∇u|p|∇u|p−2∇u+V0b|u|p|u|p−2u=f(u)+|u|q∗−1 in RNu∈W1,pRN∩W1,qRN

which the functional associated I₀ is given by

I 0 ( u ) = 1 p ∫ R N [ A ( | ∇ u | p ) + V 0 B ( | u | p ) ] d x − ∫ R N F ( u ) d x − 1 q ∗ ∫ R N | u | q ∗ d x ,

and the corresponding Nehari manifold is given by

N 0 = { u ∈ W 1 , p ( R N ) ∩ W 1 , q ( R N ) ∖ { 0 } ; I 0 ′ ( u ) u = 0 } .

We also define

c0=infN0I0.

Using the same arguments of the prove of Lemma 3.5, we conclude that

(4.1) c0<(1θ−1q∗)SN/q.

The next result allows to show that problem (P₀) has a solution that reaches c₀.

Lemma 4.1

Let (u_n) ⊂ 𝒩₀ be a sequence such that I₀(u_n)→ c₀. Then there are a sequence (y_n) ⊂ ℝ^N and constants R, η > 0 such that

(4.2) lim sup n → ∞ ∫ B R ( y n ) | u n | q d x ≥ η .

Proof. Suppose that (4.2) is not satisfied. Since (u_n) is bounded in W^1,p(ℝ^N) ∩ W^1,q(ℝ^N) we have, by in [24, Lemma 2.1],

lim n → ∞ ∫ R N | u n | s d x = 0 for all s ∈ ( q , q ∗ ) .

Hence, from (f₁)− (f₃),

∫RNf(un)undx=on(1).

Since we also have (g₃) and that Iϵn′(un)un=on (1), we get

∫ R N | u n | q ∗ d x = ∫ R N a ( | ∇ u n | p ) | ∇ u n | p d x + V 0 b ( | u n | p ) | u n | p d x + o n ( 1 ) := l

We claim that l > 0. Indeed, if the claim is not true then, by (a₁) and (b₁), we have c₀ = 0 which is a contradiction. Therefore

(4.3) lim n → ∞ ∫ R N | u n | q ∗ d x = l > 0.

By definition of the constant S, we have

(4.4) S ≤ ∫ R N | ∇ u n | q d x ∫ R N | u n | q ∗ d x q / q ∗ ≤ l q / N .

Thus, using (1.2), (1.3) and (f₃), we deduce that

c 0 + o n ( 1 ) = I 0 ( u n ) − 1 θ I 0 ( u n ) u n ≥ 1 θ − 1 q ∗ ∫ R N | u n | q ∗ d x + o n ( 1 ) .

Using (4.3), (4.4) and that c₀ > 0, we obtain c0≥1θ−1q∗SN/q which is a contradiction with (4.1).

We are going to show that the problem (P₀) has a solution that reaches the level c₀.

Lemma 4.2

(A Compactness Lemma) Let (u_n) ⊂ 𝒩₀ be a sequence satisfying I₀(u_n) → c₀. Then there exists a sequence (y˜n)⊂RN such that, up to a subsequence, vn(x)=un(x+y˜n) converges strongly in W^1,p(ℝ^N) ∩ W^1,q(ℝ^N). In particular, there exists a minimizer for c₀.

Proof. Applying Ekeland’s Variational Principle (see Theorem 8.5 in [34]),wemay suppose that (u_n) is a (PS)_c₀ for I₀. Since (u_n) is bounded in W^1,p(ℝ^N) ∩ W^1,q(ℝ^N) we can assume, up to subsequences, that u_n ⇀ u in W^1,p(ℝ^N) ∩ W^1,q(ℝ^N).

Using arguments found in [4, Lemma 2.3], we have that

(4.5) ∇un(x)→∇u(x) a.e in RNandI0′(u)=0.

Then, by (1.2), (1.3) and the Fatou’s Lemma,

0≤1p∫RNA(|∇u|p)+V0B(|u|p)dx−1θ∫RNa(|∇u|p)|∇u|p+V0B(|u|p)|u|pdx≤lim infn→+∞1p∫RNA(|∇un|p)+V0B(|un|p)dx−1θ∫RNa(|∇un|p)|∇un|p+V0B(|un|p)|un|pdx

Hence, if u ∈ 𝒩₀,

c0≤I0(u)−1θI0′(u)u≤lim infn→+∞I0(un)−1θI0′(un)un=limn→+∞I0(un)=c0.

By (4.5), (a₁), (b₁) and Lebesgue’s theorem we conclude that un→uinW1,p(RN)∩W1,q(RN). Consequently, I₀(u) = c₀ and the sequence (y˜n) is the null sequence.

If u ≡ 0, then in that case we cannot have u_n → u strongly in W^1,p(ℝ^N) ∩ W^1,q(ℝ^N) because c_V0 > 0. Hence, using Lemma 4.1, there exists a sequence {y˜n} ⊂ ℝ^N such that

vn⇀vinW1,p(RN)∩W1,q(RN),

where vn:=un(x+y˜n). Therefore, (v_n) is also a (PS)_c₀ sequence for I₀ and v/≡0. It follows from the above arguments that, up to a subsequence, (v_n) converges strongly in W^1,p(ℝ^N) ∩ W^1,q(ℝ^N) and the proof is complete.

5 Concentration results

In this section we prove some technical results in order to show the concentration result.

Proposition 5.1

Let ϵ_n → 0 and (u_n) ⊂ 𝒩_ϵn be such that I_ϵn (u_n)→ c₀. Then there exists a sequence (y˜n)⊂ ℝ^N such that vn(x)=un(x+y˜n) has a convergent subsequence in W^1,p(ℝ^N) ∩ W^1,q(ℝ^N). Moreover, up to a subsequence, y_n → y ∈ Ω, where yn=ϵny˜n.

Proof. Since V satisfies (V₁) and c₀ > 0, we repeat the same arguments in Lemma 4.1 to conclude that there exist positive constants R andβ˜ and a sequence (y˜n)⊂RN such that

lim infn→∞∫BR(y˜n)∣un∣q≥β˜>0.

Since the sequence (u_n) is bounded in W^1,p(ℝ^N) ∩ W^1,q(ℝ^N) we immediately obtain, up to a subsequence, vn⇀v/≡0 in W^1,p(ℝ^N) ∩ W^1,q(ℝ^N), where vn(x):=un(x+y˜n). Let t_n > 0 be such that

(5.1) v˜n=tnvn∈N0.

Then, since un∈Nϵn, we have

(5.2) c0≤I0(v˜n)≤Iϵn(v˜n)≤Iϵn(vn)=Iϵn(un)=c0+on(1),

which implies that I0(v˜n)→c0, as n → +∞.

From boundedness of (v_n) and (5.2), we obtain that (t_n) is bounded. As a consequence, the sequence (v˜n) is also bounded in W^1,p(ℝ^N) ∩ W^1,q(ℝ^N) which implies, up to a subsequence, v˜n⇀v˜ weakly in W^1,p(ℝ^N) ∩ W^1,q(ℝ^N).

Note that we can assume that t_n → t₀ > 0. Then, this limit implies that v˜/≡0. From Lemma 4.2, we conclude that vn˜→v˜ in W^1,p(ℝ^N) ∩ W^1,q(ℝ^N) and this implies that v_n → v in W^1,p(ℝ^N) ∩ W^1,q(ℝ^N).

To conclude the proof of this proposition, we consider yn:=ϵny˜n. Our goal is to show that (y_n) has a subsequence, still denoted by (y_n), satisfying y_n → y for y ∈ Ω. First of all, we claim that (y_n) is bounded. Indeed, suppose that there exists a subsequence, still denote by (y_n), verifying | y_n| → ∞. From(a₁), (b₁) and (V₁) we have

∫RNk1|∇vn|p+|∇vn|qdx+V0∫RNk3|vn|p+|vn|qdx≤∫RNg(ϵnx+yn,vn)vndx.

Fix R > 0 such that B_R(0) ⊃ Ω and let 𝒳_BR₍₀₎ be the characteristic function of B_R(0). Since XBR(0)(ϵx+yn)=on(1) for all x ∈ B_R(0) and v_n → v in W^1,p(ℝ^N) ∩ W^1,q(ℝ^N), then

∫RNXBR(0)(ϵx+yn)g(ϵx+yn,vn)vndx=on(1).

By definition off˜ we obtain that

∫RNk1|∇vn|p+|∇vn|qdx+V0∫RNk3|vn|p+|vn|qdx≤∫RN∖BR(0)f˜(vn)vndx+on(1)≤V0β∫RN|vn|qdx+on(1).

It follows that v_n → 0 in W^1,p(ℝ^N) ∩ W^1,q(ℝ^N), obtain this way a contradiction because c₀ > 0.

Hence (y_n) is bounded and, up to a subsequence,

yn→y‾∈RN.

Arguing as above, if yˉ∉Ωˉ we will obtain again v_n → 0 in W^1,p(ℝ^N) ∩ W^1,q(ℝ^N), and then y‾∈Ω‾. Now if V(y) = V₀, we have y‾∈/∂Ω and consequently y ∈ Ω. Suppose by contradiction that V(y‾)>V0. Then, we have

c0=I0(v˜)<1p∫RNA(|∇v˜|p)dx+1p∫RNV(y‾)B(|v˜|p)dx−∫RNF(v˜)dx−∫RN|v˜|q∗dx.

Using the fact that v˜n→v˜ in W^1,p(ℝ^N) ∩ W^1,q(ℝ^N), from Fatou’s Lemma we obtain

c0<lim infn→∞[1p∫RNA(|∇v˜n|p)dx+1p∫RNV(ϵnz+yn)B(|v˜n|p)dx−∫RNF(v˜n)dx−∫RN|v˜n|q∗dx.]

Since un∈Nϵn, this implies that

c0<lim infn→∞Iϵn(tnun)≤lim infn→∞Iϵn(un)=c0,

obtaining a contradiction.

Lemma 5.2

Let (ϵ_n) be a sequence such that ϵ_n → 0 and (un)⊂Nϵn a solution of problem (Pϵaux) . Then (v_n) converges uniformly on compacts of ℝ^N, where vn(x):=un(x+y˜n). Moreover, given ξ > 0, there exist R > 0 and n₀ ∈ N such that

∥ v n ∥ L ∞ ( R N ∖ B R ( 0 ) ) < ξ for all n ≥ n 0 ,

where (y˜n) is the sequence of Proposition 5.1.

Proof. Note that v_n is a solution of problem

−diva∇vnp∇vnp−2∇vn+Vϵx+ynbvnpvnp−2vn=gϵx+yn,vn in RN,vn∈Wϵ,

where yn=ϵny˜n. Adapting some arguments explored in [4, Lemma 5.5], we have that the sequence (v_n) is bounded in L^∞(ℝ^N) and there exist R > 0 and n₀ ∈ N such that

∥vn∥L∞(RN∖BR(0)<ξ,for alln≥n0.

Then, for any bounded domain Ω′ ⊂ ℝ^N, from (g₁)− (g₂) and continuity of V there exists C > 0 such that

|V(ϵx+yn)vnp−1−g(ϵx+yn,vn)|≤C,for alln∈N.

Hence,

|V(ϵx+yn)vnp−1−g(ϵx+yn,vn)|≤C+|∇vn|p,for alln∈N.

Considering Ψ(x) = C, we get that Ψ∈Lt(Ω′) with t>pp−1N. From [13, Theorem 1], we have

∥∇vn∥∈Lloc∞(RN).

Therefore, for all compact K ⊂ Ω^′ there exists a constant C₀ > 0, dependent only on C, N, p and dist(K,∂Ω′), such that

|∇vn|∞,K≤C0.

Then,

|vn|Cloc0,ν(RN)≤C,for alln∈Nand0<ν<1.

From Schauder’s embedding, (v_n) has a subsequence convergent in Cloc0,ν(RN).

Lemma 5.3

Given ϵ > 0, the solution u_ϵ of problem (Pϵaux) satisfies

limϵ→0Iϵ(uϵ)=cV0.

Proof. Consider z₀ ∈ Ω such that V(z₀) = V₀. Let us now consider R > 0 and set Q ∈ ∂B_R(z₀). If necessary, take R small enough such that B(Q, R/4) ⊂ Ω. Taking ψ : ℝ^N → ℝ such that ψ ≡ 1 in B(Q, R/4) and ψ ≡ 0 in ℝ^N\B(Q, R/2).

Let w₀ ∈ W^1,p(ℝ^N) ∩ W^1,q(ℝ^N) be a ground-state positive solution of the problem (P₀) which satisfies c₀ = I₀(w₀) (see Lemma 4.2). Then, we consider the function w_ϵ : ℝ^N → ℝ be given by

wϵ(x):=ψi(ϵx)w0x−z0ϵ∈Wϵ

and t_ϵ > 0, such that t_ϵw_ϵ ∈ 𝒩_ϵ. Then, with a direct computation, we have

Iϵ(uϵ)≤Iϵ(tϵwϵ)=c0+oϵ(1).

Finally, taking R → 0 in the last inequality and using the continuity of the minimax function (see [1], [31]) we get

lim supϵ→0Iϵ(uϵ)≤c0.

Let t_ϵ_,0 > 0 be such that t_ϵ_,0u_ϵ ∈ 𝒩₀. Then,

c0≤I0(tϵ,0uϵ)≤Iϵ(tϵ,0uϵ)≤Iϵ(uϵ)

and the proof is complete.

Lemma 5.4

Let (ϵ_n) be a sequence such that ϵ_n → 0 and for each n ∈ N, let (u_n) ⊂ 𝒩_ϵn be a solution of problem (Pϵaux). Then, there are δ^* > 0 and n₀ ∈ N such that, for vn(x)=un(x+y˜n), we have

vn(x)≥δ⋆,forallx∈BR(0)andn≥n0,

where R > 0 and (y˜n) were given in Lemma 5.2.

Proof. Suppose, by contradiction, that u n L ∞ ( | x | < R ) = u n L ∞ x − y ~ n < R → 0. By Lemma 5.2, we have ∥vn∥L∞(RN)→0. It follows from (f₁) that

(5.3) |f(vn)+vnq∗−1|≤V02|vn|q−1fornsufficient large.

Thus,

∫RNa(|∇vn|p)|∇vn|pdx+∫RNV(ϵnx+yn)b(|vn|p)|vn|pdx=∫RNf(vn)vndx+on(1)≤V02∫RN|vn|qdx+on(1),

which implies from (a₁) and (b₁) that,

∥un∥Wϵn→0,

which is a contradiction with Lemma 5.3.

We are now ready to show the concentration of the ground state solution.

Lemma 5.5

If Pϵϵ is the maximum point of u_ϵ, then

limϵ→0V(Pϵ)=V0.

Proof. We first notice that using Lemma 5.4 there exist δ^* > 0 and n₀ ∈ N such that

vn(qn):=maxz∈RNvn(z)=un(qn+y˜n)≥un(x)≥δ∗, for all n ≥ n₀, for all x ∈ B_R(0).

We claim that (q_n) is bounded, otherwise using Lemma 5.2 and 5.4, there exists R^* > 0 such that ∥vn∥L∞(RN∖BR∗)≤δ∗2, which implies that |vn(qn)|≤δ∗2, where we obtain a contradiction.

Then, Pϵn=ϵnqn+yn which implies

limn→+∞Pϵn=limn→+∞yn=y‾∈Ω.

Hence from continuity of V it follows that

limn→+∞V(Pϵn)=V(y‾)≥V0.

We claim that V(y‾)=V0. Indeed, suppose by contradiction that V ( y ¯ ) > V 0 . Then, we have

c0=I0(v˜)<1p∫RNA(|∇v˜|p)dx+1p∫RNV(y‾)B(|v˜|p)dx−∫RNF(v˜)−1q∗∫RN|v˜|q∗dx.

Using that v˜n→v˜inW1,p(RN)∩W1,q(RN) we obtain, from Fatou’s Lemma,

c0<lim infn→∞[1p∫RNA(|∇v˜n|p)dx+1p∫RNV(ϵnz+yn)B(|v˜n|p)dx−∫RNF(v˜n)−1q∗∫RN|v˜n|q∗dx],

and therefore

c0<lim infn→∞Iϵn(tnun)≤lim infn→∞Iϵn(un)=c0.

This contradiction shows that V(y‾)=V0.

Lemma 5.6

Let {ϵ_n} be a sequence of positive numbers such that ϵ_n → 0 as n → ∞and let (xn)⊂Ω‾ϵn be a sequence such that u_ϵn (x_n)≥ Υ > 0for some constant Υ, where for each n∈N,uϵn is a solution of (Pϵaux). Then,

limn→∞V(x‾n)=V0

where x‾n=ϵnxn.

Proof. Up to a subsequence,

x‾n→x‾∈Ω‾.

From Lemma 5.3 we have that

Iϵn(uϵn)→c0,

and there exists a positive constant C such that

∥uϵn∥≤C,∀n∈N,for someC>0.

Setting vn(z):=uϵn(z+xn),we have∥vn∥≤Candvn⇀vinW1,p(RN)∩W1,q(RN). Recalling that

vn(0)=uϵn(xn)≥Y>0,

we conclude that v/≡0.

Fix t_n > 0 verifying v˜n=tnvn∈N0, for each n ∈ N. Hence,

c0≤I0(v˜n)≤Iϵn(tnvn)≤Iϵ(vn)=Iϵ(un)=c0+on(1).

Thus, I0(v˜n)→c0, with {v˜n}⊂N0. By Lemma 4.2, we have

(5.4) v˜n→v˜inW1,p(RN)∩W1,q(RN)andI0(v˜)=c0.

Since v˜≠0, by Proposition 5.1 we have y_n = 0, for n ∈ N. Moreover, recalling that V is continuous, we have

limn→∞V(x‾n)=V(x‾).

We claim that V(x‾)=V0. Indeed, Suppose by contradiction that V(x‾)> V0, then

c0=I0(v˜)<1p∫RNA(|∇v˜|p)dx+1p∫RNV(x‾)B(|v˜|p)dx−∫RNF(v˜)dx−1q∗∫RN|v˜|q∗dx.

Thus, by (5.4) and Fatou’s Lemma, we have

c 0 < lim inf n → ∞ 1 p A ∫ R N ∇ v ~ n p d x + 1 p V ∫ R N ϵ n z + x ¯ n B v ~ n p d x − F ∫ R N v ~ n d x − 1 q ⋆ ∫ R N | v ~ | q ⋆ d x ≤ lim inf n → ∞ 1 p A ∫ R N ∇ t n v n p d x + 1 p V ∫ R N ϵ n z + x ¯ n B ∣ t n v n ∥ p d x − G ∫ R N ϵ n z + x ¯ , t n v n d x = lim inf n → ∞ I ϵ n t n u n ≤ lim inf n → ∞ I ϵ n u n = c 0 ,

which leads a absurd. Consequently limn→∞V(x‾n)=V0.

Lemma 5.7

If m_ϵ is given by mϵ=sup{max∂Ωϵuϵ: is a solution of (Pϵaux)}, then there exists ϵ‾>0 such that the sequence (m_ϵ) is bounded for all ϵ∈(0,ϵ‾). Moreover, we have limϵ→0mϵ=0.

Proof. Suppose, by contradiction, limϵ→0mϵ=+∞, then there exist u_ϵ a solution of (Pϵaux) in 𝒩_ϵ and Υ > 0 such that

max∂Ωϵuϵ≥Y>0

Thus there exists {ϵ_n}⊂ ℝ⁺ with ϵ_n → 0 and there exists a sequence {xn}⊂∂Ωϵn such that

uϵn(xn)≥Y>0.

Thus, by Lemma 5.6, we have

limn→∞V(x‾n)=V0,

where x‾n=ϵnxnand {x‾n}⊂∂Ω. Hence, up to a subsequence, we have x‾n→x‾ in ∂Ω and V(x‾)=V0, which does not make sense by (V₂). Hence, there exists ϵ > 0 such that (m_ϵ) is bounded, for all ϵ∈(0,ϵ‾).

Suppose by contradiction that there exists δ > 0 and a sequence {ϵ_n}⊂ ℝ⁺ satisfying

mϵn≥δ>0

Thus, there exists u_ϵn a solution of (P_ϵaux ) such that

mϵn−δ2<max∂Ωϵnuϵn≤mϵn.

Hence,

δ2=δ−δ2≤mϵn−δ2<max∂Ωϵuϵn,

and then there exists a sequence (x_n) ⊂ ∂Ω_ϵn , such that

uϵn(xn)≥δ2.

Repeating the above arguments, we will get an absurd. Thus, the proof is finished.

6 Proof of Theorem 1.1

Proof. Let u_ϵ be a solution of (Pϵaux). By Lemma 5.7, there exists ϵ‾>0 such that mϵ<η2 for all ϵ∈(0,ϵˉ), then (uϵ−η2)+(x)≡0 for a neighborhood from ∂Ω_ϵ. Hence, (uϵ−η2)+∈W01,p(RN∖Ωϵ)∩W01,q(RN∖Ωϵ) and the function (uϵ−η2)+∗∈W1,p(RN)∩W1,q(RN), where

(uϵ−η2)+∗(x):=0 if x∈Ωϵ,(uϵ−η2)+(x) if x∈RN∖Ωϵ.

Using (uϵ−η2)+∗ as test function. Then, by (a₁), (b₁) and (g₃)_ii, we have

0≤∫RN∖Ωϵa(|∇uϵ|p)|∇(uϵ−η2)+∗|pdx+∫RN∖Ωϵ[V0b(|uϵ|p)|uϵ|p−2−V0β|uϵ|q−2]((uϵ−η2)+∗)2dx+∫RN∖Ωϵ[V(ϵx)b(|uϵ|p)|uϵ|p−2−V0β|uϵ|q−2]η2(uϵ−η2)+∗dx=0

The last equality implies

(uϵ−η2)+∗=0,a.e inx∈RN∖Ωϵ.

This implies that |uϵ|≤η2 for z ∈ ℝ^N\Ω_ϵ, and by Remark 1 the result follows.

7 Exponential decay of the solution u_ϵ

Finally, we are going to prove the exponential decay. First technical results

Lemma 7.1

Consider M, α > 0 and ψ(x) := Mexp(−α|x|). Then

i)−div(a|∇ψ|p|∇ψ|p−2∇ψ) =αp−1−pαp+1a′(αpψp)ψ2p−1+a(αpψp)ψp−1(N−1)|x|−α(p−1),ii)−div(a|∇ψ|p|∇ψ|p−2∇ψ)≥(N−1)|x|−α(q−1)a(αpψp)αp−1ψp−1.

Proof. Note that

∂ψ∂xi(x)=Mexp(−α|x|)∂∂xi(−α|x|)=Mexp(−α|x|)(−α)xi|x|=−αxi|x|ψ(x),

which implies |∇ψ| = αψ. Then

−div(a|∇ψ|p|∇ψ|p−2∇ψ)=−∑i=1N∂∂xia|∇ψ|p|∇ψ|p−2∂ψ∂xi=αp−1∑i=1N∂∂xiaαpψpψp−1xi|x|=αp−1∑i=1Na′αpψp∂∂xiαpψpψp−1xi|x|+a(αpψp)∂∂xiψp−1xi|x|=αp−1∑i=1Na′αpψpαppψ2p−2∂ψ∂xixi|x|+a(αpψp)|x|2−xi2|x|3ψp−1+(p−1)ψp−2∂ψ∂xixi|x|=αp−1−pαp+1a′(αpψp)ψ2p−1+a(αpψp)ψp−1(N−1)|x|−α(p−1),

this prove the first item.

To prove the item ii) we are going to use (1.2) and the item i).Hence we have

−a′(αpψp)αpψp≥−(q−p)pa(αpψp),

and consequently

−pαp+1a′(αpψp)ψ2p−1≥−αψp−1(q−p)a(αpψp).

Therefore, by the item i),

−div(a|∇ψ|p|∇ψ|p−2∇ψ)≥αp−1−α(q−p)a(αpψp)ψp−1+(N−1)|x|−α(p−1)a(αpψp)ψp−1=(N−1)|x|−α(q−1)a(αpψp)αp−1ψp−1.

Corollary 7.2

Since V(x) ≥ V₀ in ℝ^N, then for α > 0 small enough we have

−div(a|∇ψ|p|∇ψ|p−2∇ψ)+k3V0ψp−1+V04ψq−1≥0 in RN.

Proof. Using (a₁) and Lemma 7.1 we obtain that

−div(a|∇ψ|p|∇ψ|p−2∇ψ)≥−α(q−1)a(αpψp)αp−1ψp−1≥−α(q−1)k2αp−1ψp−1+αq−1ψq−1=−α(q−1)k2αp−1ψp−1−α(q−1)αq−1ψq−1

Moreover, since V₀ > 0 and α > 0 is small enough, we concluded that

k3V0−α(q−1)k2αp−1≥0

and

V04−α(q−1)αq−1≥0.

Consequently

−div(a|∇ψ|p|∇ψ|p−2∇ψ)+k3V0ψp−1+V04ψq−1≥0 in RN.

Let us now relate the positive solution v_ϵ to the exponential function ψ for small ϵ.

Lemma 7.3

Let u_ϵ be the solution found in Theorem 3.1 and vϵ(x):=uϵ(x+y˜ϵ) given in Proposition 5.1. For φ_ϵ = max{v_ϵ − ψ, 0} and ϵ > 0 sufficient small, we have

∫RNa(|∇vϵ|p)|∇vϵ|p−2∇vϵ∇φϵdx+k3V0∫RN|vϵ|p−1φϵdx+V04∫RN|vϵ|q−1φϵdx≤0.

Proof. From Lemma 5.2, Lemma 5.3 and hypothesis (f₁), there exist ρ₀ > 0 such that ϵ > 0 small enough,

f(vϵ)+vϵq∗−1|vϵ|q−1≤34V0, for all|x|≥ρ0.

Since ψ(x) := Mexp(−α|x|) for x ∈ ℝ^N, we can find M˜>0 such that if M≥M˜, then φϵ:=max{|vi,ϵ|−ψ,0}≡0 in Bρ0(0) and φ_ϵ ∈ W^1,p(|x| ≥ ρ₀) ∩ W^1,q(|x| ≥ ρ₀). Therefore, the above inequality and (b₁),

∫RNa(|∇vϵ|p)|∇vϵ|p−2∇vϵ∇φϵdx+V0∫RNk3|vϵ|p−1φϵ+|vϵ|q−1φϵdx≤∫RNa(|∇vϵ|p)|∇vϵ|p−2∇vϵ∇φϵdx+∫RNV(ϵx+yϵ)b(|vϵ|p)|vϵ|p−2vϵφϵdx≤∫RNf(vϵ)φϵdx≤3V04∫RN|vϵ|q−1φϵdx

and the lemma is proved.

Finally we are going to show the exponential decay for the functions u_ϵ.

Proposition 7.4

There are ϵ₀ > 0 and C > 0 such that

|uϵ(z)|≤Cexp(−α|z−Pϵϵ|), forall z∈RN.

Proof. From [14, Lemma 2.4], we have that

a(|x|p)|x|p−2x−a(|y|p)|y|p−2y,x−y≥0, ∀ x,y∈RN.

Consider vϵ(x):=uϵ(x+y˜ϵ) the set

Λ:={x∈RN:|x|≥ρ0and|vϵ|−ψ≥0},

where ψ is the function is given by Lemma 7.1, (y˜n) is given by Proposition 5.1. Then, using Corollary 7.2 and Proposition 7.3, we obtain

0≥∫RNa(|∇vϵ|p)|∇vϵ|p−2∇vϵ−a(|∇ψ|p)|∇ψ|p−2∇ψ,∇φ˜dx+V0k3∫RN|vϵ|p−1−|ψ|p−1φ˜dx+V04∫RN|vϵ|q−1−|ψ|q−1φ˜dx≥V0k3∫RN|vϵ|p−1−|ψ|p−1φ˜dx+V04∫RN|vϵ|q−1−|ψ|q−1φ˜dx=V0k3∫Λ|vϵ|p−1−|ψ|p−1(vϵ−ψ)dx+V04∫Λ|vϵ|q−1−|ψ|q−1(vϵ−ψ)dx≥0.

Then |Λ| = 0 and consequently

vϵ(x)≤Mexp(−α|x|), ∀ |x|≥ρ0.

Considering x=z−y˜ϵ and using Lemma 5.5 there exists a constant C > 0 satisfying

(7.1) |uϵ(z)|≤Mexp−αz−yϵϵ=Mexp−αz−Pϵ+ϵqϵϵ ≤Mexp−αz−Pϵϵexp−αqϵ≤Cexp−αz−Pϵϵ,

for all |z−y˜ϵ|≥ρ0 and for ϵ > 0 small enough.

Now we are going to show the inequality (7.1) holds, for all z ∈ ℝ^N. Since (y_ϵ) converges, it follows that

|z|≥ρ0−|yϵ˜|=ρ0−|yϵ|ϵ>ρ0−1+|yϵ|ϵ→−∞ as ϵ→0.

Then, there exists ϵ₀ > 0 such that

|uϵ(z)|≤Cexp−αz−Pϵϵ, ∀ z∈RN and ∀ ϵ∈(0,ϵ0).

Appendix

In this appendix we are going to show the existence of positive solution for a problem in a bounded domain with smooth boundary, denoted by Ω. More precisely, we are going to study the following problem

−k2Δpu−Δqu+Vk4|u|p−2u+V|u|q−2u=|u|τ−2uinΩu=0on∂Ω,(Pτ)

where Ω is a bounded domain in ℝ^N and k₂, k₄, V are positive constants. We have associated to problem (P_τ) the functional

Iτ(u)=1p∫Ωk2|∇u|p+Vk4|u|pdx+1q∫Ω|∇u|q+V|u|qdx−1τ∫Ω|u|τdx

and the Nehari manifold

Nτ={u∈W01,q(Ω): u≠0 and Iτ′(u)u=0}

Lemma 7.5

For all u∈W01,q(Ω)∖{0} there exists a unique t_u ∈ (0, +∞), such that tu ∈ 𝒩_τ.

Proof. Note that if u∈W01,q(Ω)∖{0} and t > 0, we have

Iτ(tu)=tτtp−τp∫Ωk2|∇u|p+Vk4|u|pdx+tq−τq∫Ω|∇u|q+V|u|qdx−1τ∫Ω|u|τdx.

Then,

limt→0Iτ(tu)tτ=+∞ and limt→+∞Iτ(tu)tτ=−1τ∫Ω|u|τdx<0.

Consequently, there exists t_u ∈ (0, +∞) such that Iτ(tuu)=supt≥0Iτ(tu) and t_uu ∈ 𝒩_τ.

In order to show the unicity of t_u, consider f (t) = t^τ and note that f(t)tq is increasing.

Lemma 7.6

The following properties hold:

There exists ρ_τ > 0 such that ∫Ω|∇u|qdx1/q≥ρτ,forallu∈Nτ;
There exists a constant C_τ > 0 such that Iτ(u)≥Cτ∫Ω|∇u|qdx, for all u ∈ 𝒩_τ.

Proof. By Sobolev’s embeddings, there exists C > 0 such that

∫Ω|∇u|qdx≤∫Ωk2|∇u|p+Vk4|u|pdx+∫Ω|∇u|q+V|u|qdx=∫Ω|u|τdx ≤C∫Ω|∇u|qdxτ/q.

Since τ > q, the item (i) follows.

To verify the second assertion observe that

Iτ(u)=Iτ(u)−1τIτ′(u)u≥1p−1τ∫Ωk2|∇u|p+Vk4|u|pdx +1q−1τ∫Ω|∇u|q+V|u|qdx≥1q−1τ∫Ω|∇u|qdx.

Proposition 7.7

There exists wτ∈W01,q(Ω) such that w_τ is a solution of (P_τ) and I τ ( w τ ) = inf N τ I τ .

Proof. Let (u_n) be a minimizing sequence for I_τ in 𝒩_τ. By Lemma 7.6, we conclude that (u_n) is bounded in W01,q(Ω). Then there exists u∈W01,q(Ω) such that, up to a subsequence, un⇀uinW01,q(Ω) and

(7.2) un→u strongly in Ls(Ω) for any 1≤s<q∗,un(x)→u(x) for a.e x∈Ω.

Since τ ∈ (q, q^*) we have, by Lemma 7.6 again, that u ≠0. Hence,

cτ≤Iτ(tuu)≤lim infn→∞Iτ(tuun)≤lim infn→∞Iτ(un)+on(1)=cτ.

Considering w_τ := t_uu we have I_τ(w_τ) = c_τ and using Implicit Theorem we conclude that Iτ′(wτ)=0.

Acknowledgements

Gustavo S. Costa and Giovany M. Figueiredo were partially supported by CNPq, Capes and FAPDF - Brazil.

Conflict of interest: Authors state no conflict of interest.

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Received: 2021-03-07

Accepted: 2021-05-10

Published Online: 2021-07-17

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

Existence and concentration of positive solutions for a critical p&q equation

Abstract

1 Introduction

Problem 1

Problem 2

Problem 3

Problem 4

Theorem 1.1

2 Variational framework and an auxiliary problem

Remark 1

3 Existence of ground state for problem (Pϵaux)

Theorem 3.1

Lemma 3.2

Lemma 3.3

Lemma 3.4

Lemma 3.5

3.1 Proof of the Theorem 3.1

4 The Autonomous Problem

Lemma 4.1

Lemma 4.2

5 Concentration results

Proposition 5.1

Lemma 5.2

Lemma 5.3

Lemma 5.4

Lemma 5.5

Lemma 5.6

Lemma 5.7

6 Proof of Theorem 1.1

7 Exponential decay of the solution uϵ

Lemma 7.1

Corollary 7.2

Lemma 7.3

Proposition 7.4

Appendix

Lemma 7.5

Lemma 7.6

Proposition 7.7

Acknowledgements

References

Journal and Issue

Articles in the same Issue

3 Existence of ground state for problem (P_ϵaux)

7 Exponential decay of the solution u_ϵ