Sharp Blow-Up Profiles of Positive Solutions for a Class of Semilinear Elliptic Problems

Wan-Tong Li; Julián López-Gómez; Jian-Wen Sun

doi:10.1515/ans-2021-2149

Open Access Published by De Gruyter October 15, 2021

Sharp Blow-Up Profiles of Positive Solutions for a Class of Semilinear Elliptic Problems

Wan-Tong Li , Julián López-Gómez and Jian-Wen Sun

From the journal Advanced Nonlinear Studies

https://doi.org/10.1515/ans-2021-2149

Abstract

This paper analyzes the behavior of the positive solution θε of the perturbed problem

{ - Δ ⁢ u = λ ⁢ m ⁢ ( x ) ⁢ u - [ a ε ⁢ ( x ) + b ε ⁢ ( x ) ] ⁢ u p = 0 in ⁢ Ω , B ⁢ u = 0 on ⁢ ∂ ⁡ Ω ,

as ε↓0, where aε⁢(x)≈εα⁢a⁢(x) and bε⁢(x)≈εβ⁢b⁢(x) for some α≥0 and β≥0, and some Hölder continuous functions a⁢(x) and b⁢(x) such that a⪈0 (i.e., a≥0 and a≢0) and minΩ¯⁡b>0. The most intriguing and interesting case arises when a⁢(x) degenerates, in the sense that Ω0≡int⁡a-1⁢(0) is a non-empty smooth open subdomain of Ω, as in this case a “blow-up” phenomenon appears due to the spatial degeneracy of a⁢(x) for sufficiently large λ. In all these cases, the asymptotic behavior of θε will be characterized according to the several admissible values of the parameters α and β. Our study reveals that there may exist two different blow-up speeds for θε in the degenerate case.

Keywords: Reaction-Diffusion Equations; Positive Solutions; Asymptotic Profiles; Degenerate Problems

MSC 2010: 35B40; 35K57; 92D25

1 Introduction and Main Results

In this paper, we characterize the limiting behavior as ε↓0 of the classical positive solutions of the semilinear boundary value problem

(1.1) { - Δ ⁢ u = λ ⁢ m ⁢ ( x ) ⁢ u - [ a ε ⁢ ( x ) + b ε ⁢ ( x ) ] ⁢ u p = 0 in ⁢ Ω , B ⁢ u = 0 on ⁢ ∂ ⁡ Ω ,

where λ>0 is a real parameter, p>1 is constant, ε∈(0,1] is a perturbation parameter, Ω is a bounded domain of class 𝒞2+μ of ℝN (N≥1) for some μ∈(0,1), whose boundary consists of two disjoint open and closed subsets Γ0 and Γ1 of class 𝒞2+μ, ∂⁡Ω=Γ0∪Γ1, and B stands for the boundary operator defined for every u∈𝒞⁢(Γ0)⊗𝒞1⁢(Ω∪Γ1) by

(1.2) B u : = { u on ⁢ Γ 0 , ∂ ⁡ u ∂ ⁡ ν + γ ⁢ ( x ) ⁢ u on ⁢ Γ 1 .

In (1.2), ν stands for the outward unit normal to ∂⁡Ω, and γ∈𝒞μ⁢(Γ1) is an arbitrary non-negative function, γ≥0. Necessarily, Γ0 and Γ1 must possess finitely many components. As Γ0, or Γ1, can be empty, our mixed boundary operator B includes Dirichlet, Neumann and Robin boundary conditions. By simplicity, we set B≡D if Γ0=∂⁡Ω, and B≡N if Γ1=∂⁡Ω and γ=0.

A function u:Ω¯→ℝ is said to be of class 𝒞κ+μ for some integer κ≥1 if it is of class 𝒞κ⁢(Ω¯) and Dα⁢u is Hölder continuous in Ω¯, with exponent μ, for all multi-index α∈ℕN such that |α|=α1+⋯+αN=κ. A subdomain D⊂Ω is said to be of class 𝒞κ+μ if the components of its local charts in ∂⁡D are functions of class 𝒞κ+μ.

Throughout this paper, for every W∈𝒞μ⁢(Ω¯), we denote by σ1⁢[-Δ+W;B,Ω] the principal eigenvalue of the linear eigenvalue problem

{ ( - Δ + W ) ⁢ φ = σ ⁢ φ in ⁢ Ω , B ⁢ φ = 0 on ⁢ ∂ ⁡ Ω .

By applying [26, Theorem 7.10] with h≡1, it becomes apparent that σ1⁢[-Δ;B,Ω]>0 if either Γ0≠∅, or Γ0=∅ but γ⪈0. On the contrary, σ1⁢[-Δ;N,Ω]=0. Thus, under the previous general assumptions, σ1⁢[-Δ;B,Ω]≥0.

As far as m⁢(x), aε⁢(x) and mε⁢(x) are concerned, for every ε∈(0,1], they are assumed to be functions of class 𝒞μ⁢(Ω¯) such that

m ⁢ ( x + ) > 0 for some x+∈Ω,
a ε ⪈ 0 in Ω for all ε∈(0,1], and there exists a constant α≥0 such that

lim ε ↓ 0 ⁡ a ε ε α = a uniformly in ⁢ Ω ¯ ,

for some non-negative function a∈𝒞μ⁢(Ω¯) such that a-1⁢(0)=Ω¯0⊂Ω, where Ω0 is an open subdomain of class 𝒞2+μ, possibly empty,
min Ω ¯ ⁡ b ε > 0 for all ε∈(0,1], and there exists a constant β≥0 such that

lim ε ↓ 0 ⁡ b ε ε β = b uniformly in ⁢ Ω ¯ ,

for some b∈𝒞μ⁢(Ω¯) such that b⁢(x)>0 for all x∈Ω¯.

Thus, aε∼εα⁢a and bε∼εβ⁢b for small ε>0. This entails that, as soon as α>0 and β>0, the limiting problem of (1.1) as ε↓0 becomes linear. Thus, it is of a different nature than the classical problems in “homogenization theory”, where, e.g., aε⁢(x)=a⁢(ε⁢x). In this case, limε↓0⁡a⁢(ε⁢x)=a⁢(0), and hence, the limiting problem is regular if a⁢(0)>0. Our problem here is of another nature. As it will become apparent later, once having read the proofs of the main results of this paper, assumptions (Hm), (Ha) and (Hb) are optimal for their validity as they stand.

Under these general assumptions, aε⁢(x)+bε⁢(x)>0 for all x∈Ω¯ and ε∈(0,1], and hence, for every ε∈(0,1], problem (1.1) admits a unique positive solution, denoted by θε in this paper, if and only if λ>λ1B⁢(Ω) (see, e.g., Fraile et al. [15] and Aleja, Antón and López-Gómez [1]), where λ1B⁢(Ω)≥0 stands for the maximal non-negative principal eigenvalue of the weighted eigenvalue problem

(1.3) { - Δ ⁢ u = λ ⁢ m ⁢ ( x ) ⁢ u in ⁢ Ω , B ⁢ u = 0 on ⁢ ∂ ⁡ Ω .

It is well known that λ1N⁢(Ω)=0, whereas λ1B⁢(Ω)>0 if either Γ0≠∅, or Γ0=∅ but γ⪈0 (see [21, Section 1]). According to this notation, for any given subdomain Ω~ of class 𝒞μ of Ω, λ1D⁢(Ω~) stands for the principal eigenvalue of

{ - Δ ⁢ u = λ ⁢ m ⁢ ( x ) ⁢ u in ⁢ Ω ~ , u = 0 on ⁢ ∂ ⁡ Ω ~ .

Their existence goes back to Hess and Kato [19] and Brown and Lin [7] and can be derived, very easily, from the general theory of [26, Chapter 9].

The main goal of this paper is to analyze how the sharp behavior of θε as ε↓0 depends on the several possible relations between the exponents α≥0 and β≥0, which will provide us with each of the shadow limiting problems of (1.1) as ε↓0. In all cases, the sharp asymptotic profiles of θε as ε↓0 are regulated by either the positive solutions, or the minimal metasolutions in Ω∖Ω¯0, of a semilinear elliptic boundary value problem of the form

(1.4) { - Δ ⁢ u = λ ⁢ m ⁢ ( x ) ⁢ u - c ⁢ ( x ) ⁢ u p ⁢ ( x ) = 0 in ⁢ Ω , B ⁢ u = 0 on ⁢ ∂ ⁡ Ω ,

where the concrete value of c∈{a,b,a+b} is determined by the values of α and β; the precise concept of metasolution has been discussed in the monograph [27]. Note that (1.4) is non-degenerate, unless c=a and Ω0≠∅, because we are assuming that b⁢(x)>0 for all x∈Ω¯ and a+b⪈b. Throughout this paper, for every c∈𝒞μ⁢(Ω¯), c⪈0, we will denote by Uc the unique positive solution of (1.4) if it exists. According to Fraile et al. [15], Uc exists if and only if λ>λ1B⁢(Ω) unless c=a and Ω0≠∅. In the degenerate case when c=a and Ω0≠∅, problem (1.4) possesses a positive solution if and only if

(1.5) λ 1 B ⁢ ( Ω ) < λ < λ 1 D ⁢ ( Ω 0 ) .

Moreover, it is unique if it exists (see Fraile et al. [15] and Aleja, Antón and López-Gómez [1]). By adopting the notation λ1D⁢(Ω0)=+∞ if Ω0=∅, (1.5) characterizes the existence of a positive solution in all these cases. This convention is consistent with the property that

lim | Ω ~ | ↓ 0 ⁡ λ 1 D ⁢ ( Ω ~ ) = + ∞ ,

where |Ω~| stands for the Lebesgue measure of Ω~, which can be easily derived from the iso-perimetric inequality of Faber [14] and Krahn [20] (see [22, Theorem 5.1]).

The dynamics of the positive solutions of the parabolic counterpart of (1.4) are governed by the maximal non-negative solution of (1.4) if c∈{b,a+b}, or c=a with Ω0 empty, but it is regulated by its minimal positive metasolution when c=a, Ω0 is non-empty and λ≥λ1D⁢(Ω0). Thus, if the coefficient a⁢(x) has a spatial degeneracy in Ω0, i.e., Ω0≠∅, then the asymptotic behavior of the positive solutions to problem (1.1) can be quite different from the behavior of the classical perturbation problem when a⁢(x) admits no spatial degeneracy.

The analysis of these degenerate problems goes back to Brézis and Oswald [6], by means of variational techniques, Ouyang [32], through global continuation, and Fraile et al. [15], where the method of sub and supersolutions was incorporated to the theory of singular problems, and the problem of the dynamics of the parabolic counterparts of these degenerate elliptic models was addressed by the first time. Then, based on [15], the theory advanced very quickly with García-Melián et al. [16], Gómez-Reñasco and López-Gómez [17], and Du and Huang [13] (see [27] for further information).

The spatial degeneracy makes a fundamental change on the behavior of the positive solutions of (1.4) when c=a and Ω0≠∅ (see, e.g., López-Gómez [23] and Du [12] for the simplest case when m≡1). Assume that m is sufficiently smooth in Ω0 with m⁢(x)>0 for all x∈∂⁡Ω0 and m≥0 in Ω¯0. Then, by a recent result of the authors [21, Theorem 1.1], Ua satisfies

lim λ ↑ λ 1 D ⁢ ( Ω 0 ) ⁡ U a = { + ∞ uniformly in ⁢ Ω ¯ 0 , L min in ⁢ Ω ¯ ∖ Ω ¯ 0 ,

where Lmin stands for the minimal positive solution of the singular problem

(1.6) { - Δ ⁢ u = λ 1 D ⁢ ( Ω 0 ) ⁢ m ⁢ ( x ) ⁢ u - a ⁢ ( x ) ⁢ u p ⁢ ( x ) in ⁢ Ω ∖ Ω ¯ 0 , B ⁢ u = 0 on ⁢ ∂ ⁡ Ω , u = + ∞ on ⁢ ∂ ⁡ Ω 0 .

The existence of Lmin for (1.6) was established by [21, Theorem 2.1]. The “blow-up” of Ua in Ω¯0 as λ↑λ1D⁢(Ω0) appears due to the spatial degeneracy of a⁢(x) in Ω0.

On the other hand, if c=a≡0 in Ω¯, then (1.4) reduces to the linear eigenvalue problem (1.3). In this case, the sharp changes of the positive solutions between the nonlinear problem (1.4) with c=a and the linear problem (1.3) have been investigated by Sun in [33], where it was established that, whenever α≥0,

lim ε ↓ 0 ⁡ U a ε = + ∞ locally uniformly in ⁢ Ω .

These two basic results tell us that the “blow-up” phenomenon occurs when either a⁢(x) exhibits some degeneration or the whole nonlinear problem degenerates to the linear one. At the end of the day, the second situation is a limiting case of the former one when Ω0↑Ω. Our interest in analyzing those sharp changes explains why we have chosen the perturbed model (1.1), where α≥0 is the quenching speed of the nonlinear function and β≥0 is the speed of the spatial degeneracy.

Problem (1.4) is a basic semilinear elliptic boundary value problem used in the study of a variety of phenomena in the applied sciences (see, e.g., [18, 32, 5, 6, 15, 17, 10, 21, 28, 3]). It is also the most paradigmatic model in population dynamics, the diffusive logistic model [4, 9, 31, 2, 11, 17, 13, 23, 29]. In this context, Ω is the region inhabited by the population with species density u, λ⁢m⁢(x) measures the birth or death rates in a heterogenous environment, according to whether it is positive or negative, and c⁢(x) measures the carrying capacity of Ω, i.e., the capacity of Ω to support the species u⁢(x). From a biological point of view, the region {x∈Ω:m⁢(x)>0} acts like a source for u, whereas {x∈Ω:m⁢(x)<0} can be thought as a sink. The case when the growth function m⁢(x) changes sign seems to be far more interesting from the point of view of the applications than the case when m⁢(x) is everywhere positive, besides much harder to deal with from a mathematical point of view. Under the above assumptions, the semilinear problems (1.4) was well studied (see [15, 18, 31, 29, 1, 21] and the references therein).

We now deliver the main results of this paper. The first one provides us with the sharp profile of θε as ε↓0 when α=β≥0. It can be stated as follows.

Theorem 1.1.

Suppose (Hm), (Ha) and (Hb), with α=β≥0, and λ>λ1B⁢(Ω). Then

(1.7) lim ε ↓ 0 + ⁡ ( ε α p - 1 ⁢ θ ε ) = U a + b uniformly in ⁢ Ω ¯ .

In particular, limε↓0⁡θε=Ua+b uniformly in Ω¯ if α=0, while limε↓0⁡θε=+∞ locally uniformly in Ω if α>0, which covers the result of Sun [33], because limε↓0⁡(aε+bε)=0 in Ω if α=β>0. Moreover, by (1.7), when α=β>0, the positive solutions of (1.1) have a blow-up rate ε-αp-1 because θε∼ε-αp-1⁢Ua+b as ε↓0.

Our second result provides us with the sharp profile of θε as ε↓0 when α>β≥0.

Theorem 1.2.

Suppose (Hm), (Ha) and (Hb), with α>β≥0, and λ>λ1B⁢(Ω). Then

(1.8) lim ε ↓ 0 + ⁡ ( ε β p - 1 ⁢ θ ε ) = U b uniformly in ⁢ Ω ¯ .

In particular, limε↓0⁡θε=Ub uniformly in Ω¯ if β=0, while limε↓0⁡θε=+∞ locally uniformly in Ω if β>0. Moreover, when α>β>0, the positive solutions of (1.1) have a blow-up rate ε-βp-1 because, due to (1.8),

θ ε ∼ ε - β p - 1 ⁢ U b as ⁢ ε ↓ 0 .

The most intriguing case arises when β>α≥0, where the asymptotic profiles of θε as ε↓0 make a sharp change with respect to the previous ones. Our main result in this case can be stated as follows. We are denoting by r:ℕ→ℕ∪{0} the function defined by r⁢(1)=0, r⁢(2)=r⁢(3)=r⁢(4)=r⁢(5)=1, and, for every n≥2,

r ⁢ ( 4 ⁢ n - 2 ) = r ⁢ ( 4 ⁢ n - 1 ) = r ⁢ ( 4 ⁢ n ) = r ⁢ ( 4 ⁢ n + 1 ) = r ⁢ ( 4 ⁢ ( n - 1 ) ) + 2 .

It was introduced by the authors in [21].

Theorem 1.3.

Suppose (Hm), (Ha) and (Hb), with β>α≥0, m≥0 in Ω0, m⁢(x)>0 for all x∈∂⁡Ω0, and m∈Cr⁢(N)⁢(Ω0), where N is the spatial dimension. Then

when λ 1 B ⁢ ( Ω ) < λ < λ 1 D ⁢ ( Ω 0 ) ,

(1.9) lim ε ↓ 0 ⁡ ( ε α p - 1 ⁢ θ ε ) = U a uniformly in ⁢ Ω ¯ .
When λ ≥ λ 1 D ⁢ ( Ω 0 ) ,

(1.10) lim ε ↓ 0 ⁡ ( ε α p - 1 ⁢ θ ε ) = + ∞ uniformly in ⁢ Ω ¯ 0 .

Moreover, for every sequence { ε n } n ≥ 1 , with lim n → ∞ ⁡ ε n = 0 , there exists a subsequence of { θ ε n } n ≥ 1 , relabeled in the same way, such that

(1.11) lim n → ∞ ⁡ ( ε n α p - 1 ⁢ θ ε n ) = L 𝑖𝑛 ⁢ Ω ¯ ∖ Ω ¯ 0 ,

where L is a positive solution of the singular problem

(1.12) { - Δ ⁢ u = λ ⁢ m ⁢ u - a ⁢ u p 𝑖𝑛 ⁢ Ω ∖ Ω ¯ 0 , B ⁢ u = 0 𝑜𝑛 ⁢ ∂ ⁡ Ω , u = + ∞ 𝑜𝑛 ⁢ ∂ ⁡ Ω 0 .

Furthermore, if, in addition,

(1.13) lim ε ↓ 0 ⁡ a ε ε β = 0 uniformly in ⁢ Ω ¯ 0 ,

a ε + b ε ≥ ε α ⁢ a for sufficiently small ε > 0 , and the minimal positive solution L min of problem ( 1.12 ) satisfies L min ∈ L 2 ⁢ p ⁢ ( Ω ∖ Ω ¯ 0 ) , then L = L min and

(1.14) lim ε ↓ 0 ⁡ ( ε β p - 1 ⁢ θ ε ) = U b , Ω 0 uniformly in ⁢ Ω ¯ 0 ,

where U b , Ω 0 stands for the unique positive solution of

(1.15) { - Δ ⁢ u = λ ⁢ m ⁢ u - b ⁢ u p 𝑖𝑛 ⁢ Ω 0 , u = 0 𝑜𝑛 ⁢ ∂ ⁡ Ω 0 .

Note that (1.14) provides us with the exact blow-up rate of θε in Ω0 as ε↓0. More precisely, according to (1.11),

θ ε ∼ ε - α p - 1 ⁢ L min in ⁢ Ω ¯ ∖ Ω ¯ 0 as ⁢ ε ↓ 0 ,

whereas, thanks to (1.14),

θ ε ∼ ε - β p - 1 ⁢ U b , Ω 0 in ⁢ Ω ¯ 0 as ⁢ ε ↓ 0 .

Thus, θε has two different blow-up rates in Ω as ε↓0 if β>α>0.

Unless the singular problem (1.12) admits a unique positive solution, or aε+bε≥εα⁢a for sufficiently small ε>0, one cannot guarantee that, along two different subsequences of θε, the same positive solution of (1.12) will be approximated. In general, the limit might depend on the decay of aε+bε to 0 as ε↓0. Some very general conditions on a⁢(x) near ∂⁡Ω0 so that (1.12) can admit a unique large positive solution were given in [24, 25, 30] as well as in the references therein. According to [24, Theorem 1.1], this occurs, e.g., when there are two continuous functions E,C:∂⁡Ω0→(0,∞) such that, for every x0∈∂⁡Ω0,

(1.16) lim x ∈ Ω ∖ Ω ¯ 0 x → x 0 ⁡ a ⁢ ( x ) [ dist ⁡ ( x , ∂ ⁡ Ω 0 ) ] E ⁢ ( x 0 ) = C ⁢ ( x 0 ) ,

though a⁢(x) can decay to zero on ∂⁡Ω0 with rather general rates (see [25] for further information). Moreover, under condition (1.16), the unique positive solution L of (1.12) satisfies, for every x0∈∂⁡Ω0,

L ⁢ ( x ) ≈ [ 1 C ⁢ ( x 0 ) ⁢ E ⁢ ( x 0 ) + 2 p - 1 ⁢ ( E ⁢ ( x 0 ) + 2 p - 1 + 1 ) ] 1 p - 1 ⁢ [ dist ⁡ ( x , ∂ ⁡ Ω 0 ) ] - E ⁢ ( x 0 ) + 2 p - 1

when x∈Ω∖Ω¯0 approximates x0. Thus, in the special but important case when the decay rate of a⁢(x) to zero on ∂⁡Ω0, measured by E, is constant on ∂⁡Ω0, i.e., there exists a constant q>0 such that E⁢(x0)=q for all x0∈∂⁡Ω0, then L∈L2⁢p⁢(Ω∖Ω¯0) if and only if

(1.17) ∫ Ω ∖ Ω ¯ 0 [ dist ⁡ ( x , ∂ ⁡ Ω 0 ) ] - 2 ⁢ p ⁢ q + 2 p - 1 ⁢ 𝑑 x < + ∞ .

It is folklore that (1.17) holds true if and only if N>2⁢p⁢q+2p-1. Therefore, in this case, all the assumptions of the last part of Theorem 1.3 are satisfied for sufficiently large N≥1. The challenge of analyzing whether or not the assumption L∈L2⁢p⁢(Ω∖Ω¯0) can be removed from the statement of Theorem 1.3 remains an open problem in this paper.

Finally, note that, as soon as β>α>0, one has that

a ε ε β = ε α - β ⁢ a ε ε α

with α-β<0. Thus, (Ha) and (1.13) entail that

lim ε ↓ 0 ⁡ a ε ε α = a ≡ 0 in ⁢ Ω ¯ 0

at a rate higher than εβ-α.

The rest of this paper is organized as follows. Section 2 delivers the proof of a general comparison result, Lemma 2.1, which is used throughout this paper, Section 3 gives the proofs of Theorems 1.1 and 1.2, and Section 4 consists of the proof of Theorem 1.3.

2 A Useful Comparison Result

This section studies an extremely useful comparison result, whose proof will be delivered by the sake of completeness.

Lemma 2.1.

Let c⪈0 be a function of class Cμ⁢(Ω¯) such that problem (1.4) possesses a positive solution Uc, necessarily unique by, e.g., [1, Theorem 3.1]. Then, for every positive strict subsolution u¯ of (1.4) such that u¯⁢(x)>0 for all x∈Ω, one has that Uc-u¯≫0 in the sense that Uc⁢(x)-u¯⁢(x)>0 for all x∈Ω, and ∂ν⁡(Uc-u¯)⁡(x)<0 for all x∈∂⁡Ω such that u¯⁢(x)=Ua⁢(x). Similarly, for every positive strict supersolution u¯ of (1.4) such that u¯⁢(x)>0 for all x∈Ω, one has that u¯-Uc≫0.

Proof.

Since Uc is a positive solution of (1.4), we have that

{ ( - Δ + c ⁢ U c p - 1 - λ ⁢ m ) ⁢ U c = 0 in ⁢ Ω , B ⁢ u = 0 on ⁢ ∂ ⁡ Ω .

Thus, according to [26, Theorem 7.7],

(2.1) σ 1 ⁢ [ - Δ + c ⁢ U c p - 1 - λ ⁢ m ; B , Ω ] = 0 .

Let u¯ be a positive strict subsolution of (1.4). By definition,

{ - Δ ⁢ u ¯ ≤ λ ⁢ m ⁢ ( x ) ⁢ u ¯ - c ⁢ ( x ) ⁢ u ¯ p in ⁢ Ω , B ⁢ u ¯ ≤ 0 on ⁢ ∂ ⁡ Ω ,

with some of these inequalities strict. Thus,

(2.2) B ⁢ ( U c - u ¯ ) ≥ 0 on ⁢ ∂ ⁡ Ω ,

and

- Δ ⁢ ( U c - u ¯ ) ≥ λ ⁢ m ⁢ ( U c - u ¯ ) - c ⁢ ( U c p - u ¯ p ) = λ ⁢ m ⁢ ( U c - u ¯ ) - p ⁢ c ⁢ ∫ 0 1 ( t ⁢ U c + ( 1 - t ) ⁢ u ¯ ) p - 1 ⁢ 𝑑 t ⁢ ( U c - u ¯ ) ,

with some of these inequalities strict. Equivalently,

(2.3) ( - Δ + V - λ ⁢ m ) ⁢ ( U c - u ¯ ) ≥ 0 in ⁢ Ω ,

where

V ≡ p ⁢ c ⁢ ∫ 0 1 ( t ⁢ U c + ( 1 - t ) ⁢ u ¯ ) p - 1 ⁢ 𝑑 t .

Since u¯⁢(x)>0 for all x∈Ω and c⪈0, it becomes apparent that

p ⁢ ∫ 0 1 ( t ⁢ U c ⁢ ( x ) + ( 1 - t ) ⁢ u ¯ ⁢ ( x ) ) p - 1 ⁢ 𝑑 t > U c p - 1 ⁢ ( x ) for all ⁢ x ∈ Ω ,

and hence, V⪈c⁢Ucp-1 in Ω. Consequently, by [8, Proposition 3.3], we find that

σ 1 ⁢ [ - Δ + V - λ ⁢ m ; B , Ω ] > σ 1 ⁢ [ - Δ + c ⁢ U c p - 1 - λ ⁢ m ; B , Ω ] = 0 .

Therefore, since either (2.2) or (2.3) is strict, we can conclude from [26, Theorem 7.10] that Ua-u¯≫0.

The second assertion can be easily accomplished by adapting the previous argument. So we will omit any further technical details herein. ∎

3 Proof of Theorems 1.1 and 1.2

The proof of these theorems relies on the next (regular) perturbation result.

Theorem 3.1.

Suppose (Hm) and cε∈Cμ⁢(Ω¯), ε∈[0,1], satisfies c0⁢(x)>0 for all x∈Ω¯⁢m⁢e⁢g⁢a and

(3.1) lim ε ↓ 0 ⁡ c ε = c 0 uniformly in ⁢ Ω ¯ .

Then

(3.2) lim ε ↓ 0 ⁡ U c ε = U c 0 uniformly in ⁢ Ω ¯ .

Proof.

By (3.1), for every η>0, there exists ε0=ε0⁢(η) such that

c 0 - η ≤ c ε ≤ c 0 + η for all ⁢ ε ∈ ( 0 , ε 0 ) .

Hence, for every ε∈(0,ε0), we have that

λ ⁢ m ⁢ U c ε - ( c 0 + η ) ⁢ U c ε p ≤ λ ⁢ m ⁢ U c ε - c ε ⁢ U c ε p = - Δ ⁢ U c ε ≤ λ ⁢ m ⁢ U c ε - ( c 0 - η ) ⁢ U c ε p .

Consequently, for every ε∈(0,ε0), Ucε is a subsolution of

{ - Δ ⁢ u = λ ⁢ m ⁢ u - ( c 0 - η ) ⁢ u p in ⁢ Ω , B ⁢ u = 0 on ⁢ ∂ ⁡ Ω ,

and, simultaneously, it is a supersolution of

(3.3) { - Δ ⁢ u = λ ⁢ m ⁢ u - ( c 0 + η ) ⁢ u p in ⁢ Ω , B ⁢ u = 0 on ⁢ ∂ ⁡ Ω .

Since c0⁢(x)>0 for all x∈Ω¯, η>0 can be shortened, if necessary, so that c0⁢(x)-η>0 for all x∈Ω¯. So the positive solutions Uc0±η are well defined because we are assuming that λ>λ1B⁢(Ω). Moreover, thanks to Lemma 2.1, they satisfy

U c 0 + η ≤ U c ε ≤ U c 0 - η in ⁢ Ω for all ⁢ ε ∈ ( 0 , ε 0 ) .

Therefore, (3.2) holds true as soon as

(3.4) lim η ↓ 0 ⁡ U c 0 ± η = U c 0 uniformly in ⁢ Ω ¯ ,

whose validity can be easily shown, e.g., by applying the implicit function theorem to the pair (η,u)=(0,Uc0) viewed as a solution pair of, e.g., (3.3). Indeed, the solutions of (3.3) are given, e.g., by the zeroes of the operator 𝔉:ℝ×𝒞B1+μ⁢(Ω¯)→𝒞B1+μ⁢(Ω¯) defined by

𝔉 ⁢ ( ξ , u ) = u - ( - Δ + 1 ) - 1 ⁢ [ ( λ ⁢ m + 1 ) ⁢ u - ( c 0 + ξ ) ⁢ u p ] , ( ξ , u ) ∈ ℝ × 𝒞 B 1 + μ ⁢ ( Ω ¯ ) ,

where (-Δ+1)-1 stands for the resolvent operator of -Δ+1 in Ω subject to the boundary operator B, and

𝒞 B 1 + μ ⁢ ( Ω ¯ ) ≡ { u ∈ 𝒞 1 + μ ⁢ ( Ω ¯ ) : B ⁢ u = 0 ⁢ on ⁢ ∂ ⁡ Ω } .

By definition, 𝔉⁢(0,Uc0)=0. Moreover, since p>1, 𝔉 is of class 𝒞1 and

D u ⁢ 𝔉 ⁢ ( 0 , U c 0 ) ⁢ u = u - ( - Δ + 1 ) - 1 ⁢ [ ( λ ⁢ m + 1 ) ⁢ u - p ⁢ c 0 ⁢ U c 0 p - 1 ⁢ u ]

for all u∈𝒞B1+μ⁢(Ω¯). Since it is a compact perturbation of the identity map, Du⁢𝔉⁢(0,Uc0) is a Fredholm operator of index zero. Thus, it is an isomorphism if ker⁡Du⁢𝔉⁢(0,Uc0)=[0]. On the contrary, assume that Du⁢𝔉⁢(0,Uc0)⁢u=0 for some u∈𝒞B1+μ⁢(Ω¯), u≠0. Then, by elliptic regularity, u∈𝒞B2+μ⁢(Ω¯) and

( - Δ + p ⁢ c 0 ⁢ U c 0 p - 1 - λ ⁢ m ) ⁢ u = 0 in ⁢ Ω .

Thus, σ1⁢[-Δ+p⁢c0⁢Uc0p-1-λ⁢m;B,Ω]≤0. Hence, by [8, Proposition 3.3],

0 ≥ σ 1 ⁢ [ - Δ + p ⁢ c 0 ⁢ U c 0 p - 1 - λ ⁢ m ; B , Ω ] > σ 1 ⁢ [ - Δ + c 0 ⁢ U c 0 p - 1 - λ ⁢ m ; B , Ω ] ,

which contradicts (2.1) and shows that Du⁢𝔉⁢(0,Uc0) indeed establishes an isomorphism. Therefore, by the implicit function theorem, there exist η0>0 and a map of class 𝒞1, u:(-η0,η0)→𝒞B1+μ⁢(Ω¯), such that u⁢(0)=Uc0 and 𝔉⁢(η,u⁢(η))=0 for all η∈(-η0,η0). Since Uc0≫0, it is apparent that u⁢(η)≫0 for sufficiently small η. Consequently, by the uniqueness of the positive solution, we find that u⁢(η)=Uc0+η. By the continuity of u⁢(η), this shows (3.4) and ends the proof. ∎

3.1 Proof of Theorem 1.1

Suppose that α=β=0. Then limε↓0⁡(aε+bε)=a+b uniformly in Ω¯, and hence, Theorem 1.1 is a direct consequence of Theorem 3.1.

Suppose that α=β>0. Then the positive function ωε≡εαp-1⁢θε solves the problem

{ - Δ ⁢ v = λ ⁢ m ⁢ ( x ) ⁢ v - ( a ε ε α + b ε ε α ) ⁢ v p in ⁢ Ω , B ⁢ v = 0 on ⁢ ∂ ⁡ Ω .

Thus, since

lim ε ↓ 0 ⁡ ( a ε ε α + b ε ε α ) = a + b uniformly in ⁢ Ω ¯ ,

we can infer from Theorem 3.1 that limε↓0⁡ωε=Ua+b. The proof is complete.

3.2 Proof of Theorem 1.2

Suppose that α>β=0. Then limε↓0⁡(aε+bε)=b uniformly in Ω¯, and, also in this case, Theorem 1.2 follows as a direct consequence of Theorem 3.1.

Suppose that α>β>0. Then the positive function ωε≡εβp-1⁢θε solves the problem

{ - Δ ⁢ v = λ ⁢ m ⁢ ( x ) ⁢ v - ( a ε ε β + b ε ε β ) ⁢ v p in ⁢ Ω , B ⁢ v = 0 on ⁢ ∂ ⁡ Ω .

Since α>β, it follows from (Ha) and (Hb) that

lim ε ↓ 0 ⁡ ( a ε ε β + b ε ε β ) = lim ε ↓ 0 ⁡ ( ε α - β ⁢ a ε ε α + b ε ε β ) = b uniformly in ⁢ Ω ¯ .

Therefore, also by Theorem 3.1, limε↓0⁡ωε=Ub, which ends the proof of Theorem 1.2.

4 Proof of Theorem 1.3

As Theorem 1.3 is a very deep result, its proof is substantially more sophisticated than the proofs of Theorems 1.1 and 1.2. We will divide it into four steps.

Step 1: Proof of (1.9) for β>α=0 and λ1B⁢(Ω)<λ<λ1D⁢(Ω0). Suppose β>α=0. Then, by (Ha) and (Hb),

(4.1) lim ε ↓ 0 ⁡ ( a ε + b ε ) = a uniformly in ⁢ Ω ¯ .

Suppose, in addition, that λ1B⁢(Ω)<λ<λ1D⁢(Ω0). By [21, Theorem 1.2], the positive solution Ua is well defined and it is unique. Note that, in case α=0, (1.9) simplifies to

(4.2) lim ε ↓ 0 ⁡ θ ε = U a uniformly in ⁢ Ω ¯ .

By (4.1), we obtain that, for sufficiently small ε>0, aε≤aε+bε≤a⁢(x)+1 for all x∈Ω¯. Subsequently, we will assume that ε has been chosen in this way. Then

λ ⁢ m ⁢ θ ε - ( a + 1 ) ⁢ θ ε p ≤ λ ⁢ m ⁢ θ ε - ( a ε + b ε ) ⁢ θ ε p = - Δ ⁢ θ ε ≤ λ ⁢ m ⁢ θ ε - a ε ⁢ θ ε p

and hence, by Lemma 2.1,

(4.3) U a + 1 ≤ θ ε ≤ U a ε in ⁢ Ω .

Subsequently, for sufficiently small δ>0, we consider the open subset of Ω defined by

Ω δ ≡ { x ∈ Ω : dist ⁡ ( x , Ω 0 ) < δ } .

By construction, Ω¯0⊂Ωδ and, since m≥0 in Ωδ for sufficiently small δ>0, using the general theory of [26, Chapter 9], it is apparent that δ can be shortened so that

(4.4) λ < λ 1 D ⁢ ( Ω δ ) < λ 1 D ⁢ ( Ω 0 ) .

Pick any function a~δ∈𝒞μ⁢(Ω¯) such that a~δ-1⁢(0)=Ω¯δ and a~δ⁢(x)<a⁢(x) for all x∈Ω¯∖Ω¯0. Then, since aε→a as ε↓0 uniformly in Ω¯ and aε⁢(x)>0 for all x∈Ω¯, one has that a~δ≤aε for sufficiently small ε>0. Thus,

- Δ ⁢ U a ε = λ ⁢ m ⁢ U a ε - a ε ⁢ U a ε p ≤ λ ⁢ m ⁢ U a ε - a ~ δ ⁢ U a ε p

and hence, by Lemma 2.1, we find that Uaε≤Ua~δ for sufficiently small ε>0. Therefore, by (4.3), we can infer that

(4.5) U a + 1 ≤ θ ε ≤ U a ~ δ in ⁢ Ω .

For sufficiently small ε>0, εε satisfies the integral equation

(4.6) θ ε = ( - Δ + 1 ) - 1 ⁢ [ ( λ ⁢ m + 1 ) ⁢ θ ε - ( a ε + b ε ) ⁢ θ ε p ] = ( - Δ + 1 ) - 1 ⁢ [ ( λ ⁢ m + 1 ) ⁢ θ ε - a ⁢ θ ε p ] + ( - Δ + 1 ) - 1 ⁢ [ ( a - a ε - b ε ) ⁢ θ ε p ] .

Moreover, by (4.1), we have that

lim ε ↓ 0 ⁡ ( - Δ + 1 ) - 1 ⁢ [ ( a - a ε - b ε ) ⁢ θ ε p ] = 0 in ⁢ 𝒞 2 + μ ⁢ ( Ω ¯ ) .

Thus, by the compactness of the resolvent operator, it follows from (4.5) that there exists a sequence {εn}n≥1, with limn→∞⁡εn=0, such that limn→∞⁡θεn=θ0∈𝒞2+μ⁢(Ω¯). Particularizing (4.6) at ε=εn and letting n→∞ yields to θ0=(-Δ+1)-1⁢[(λ⁢m+1)⁢θ0-a⁢θ0p], i.e., θ0 solves the problem

{ - Δ ⁢ u = λ ⁢ m ⁢ ( x ) ⁢ u - a ⁢ ( x ) ⁢ u p ⁢ ( x ) in ⁢ Ω , B ⁢ u = 0 on ⁢ ∂ ⁡ Ω .

Moreover, by (4.5), θ0≫0. Therefore, by uniqueness, θ0=Ua. As this argument can be repeated along any sequence of ε’s converging to zero, (4.2) holds.

Step 2: Proof of (1.10) and (1.11) for β>α=0 and λ≥λ1D⁢(Ω0). Assume that β>α=0 and λ≥λ1D⁢(Ω0). As in Step 1, (4.1) holds. Pick a sufficiently small δ>0 such that m≥0 in Ωδ. Since θε⁢(x)>0 for all x∈∂⁡Ωδ⊂Ω, θε provides us with a positive strict supersolution of the boundary value problem

(4.7) { - Δ ⁢ u = λ ⁢ m ⁢ ( x ) ⁢ u - ( a ε + b ε ) ⁢ u p in ⁢ Ω δ , u = 0 on ⁢ ∂ ⁡ Ω δ .

According to (4.4), we have that λ≥λ1D⁢(Ω0)>λ1D⁢(Ωδ). Moreover, aε⁢(x)+bε⁢(x)>0 for all x∈Ω¯δ. Thus, (4.7) admits a unique positive solution, denoted by θ[λ,ε,δ] (see, e.g., [1] if necessary). By Lemma 2.1,

(4.8) θ [ λ , ε , δ ] ≤ θ ε in ⁢ Ω δ .

Similarly, since m≥0 in Ωδ, for sufficiently small η>0, the function θ[λ,ε,δ] provides us with a positive supersolution of

(4.9) { - Δ ⁢ u = ( λ 1 D ⁢ ( Ω 0 ) - η ) ⁢ m ⁢ ( x ) ⁢ u - ( a ε + b ε ) ⁢ u p in ⁢ Ω δ , u = 0 on ⁢ ∂ ⁡ Ω δ .

Hence, again by Lemma 2.1, it follows from (4.8) that

(4.10) θ [ λ 1 D ⁢ ( Ω 0 ) - η , ε , δ ] ≤ θ [ λ , ε , δ ] ≤ θ ε in ⁢ Ω δ .

As for sufficiently small η>0 we have that λ1B⁢(Ω)<λ1D⁢(Ω0)-η<λ1D⁢(Ω0), applying the result already proven in Step 1 to problem (4.9), it is apparent that

lim ε ↓ 0 ⁡ θ [ λ 1 D ⁢ ( Ω 0 ) - η , ε , δ ] = θ [ λ 1 D ⁢ ( Ω 0 ) - η , a , δ ] ,

where θ[λ1D⁢(Ω0)-η,a,δ] denotes the unique positive solution of

{ - Δ ⁢ u = ( λ 1 D ⁢ ( Ω 0 ) - η ) ⁢ m ⁢ ( x ) ⁢ u - a ⁢ u p in ⁢ Ω δ , u = 0 on ⁢ ∂ ⁡ Ω δ .

Consequently, letting ε↓0 in (4.10), we obtain that, for sufficiently small η,

(4.11) θ [ λ 1 D ⁢ ( Ω 0 ) - η , a , δ ] ≤ lim inf ε ↓ 0 ⁡ θ ε in ⁢ Ω δ .

Since under the general assumptions of Theorem 3.1, [21, Theorem 1.1] implies that

(4.12) lim η ↓ 0 ⁡ θ [ λ 1 D ⁢ ( Ω 0 ) - η , a , δ ] = + ∞ uniformly in ⁢ Ω ¯ 0 = a - 1 ⁢ ( 0 ) ,

letting η↓0 in (4.11), we can conclude from (4.12) that lim infε↓0⁡θε=+∞ uniformly in Ω¯0. This ends the proof of (1.10) for α=0.

It remains to prove the second assertion of part (b) for α=0. By (Ha) and (Hb), there exists δ0>0 such that, for every δ∈(0,δ0), there are ε0=ε0⁢(δ)>0 and η0=η0⁢(δ)>0 such that

a ε + b ε ≥ a - η ≥ min Ω ¯ ∖ Ω δ ⁡ a - η > 0 in ⁢ D δ ≡ Ω ∖ Ω ¯ δ

for all ε∈(0,ε0) and η∈(0,η0). Thus, thanks to Lemma 2.1,

(4.13) θ ε ≤ L a - η , D δ min in ⁢ D δ ≡ Ω ∖ Ω ¯ δ for every ⁢ ε ∈ ( 0 , ε 0 ) ,

where La-η,Dδmin stands for the minimal positive solution of the singular problem

{ - Δ ⁢ u = λ ⁢ m ⁢ u - ( a - η ) ⁢ u p ⁢ ( x ) in ⁢ D δ , B ⁢ u = 0 on ⁢ ∂ ⁡ Ω , u = + ∞ on ⁢ ∂ ⁡ Ω δ .

Observe that ∂⁡Dδ=∂⁡Ω∪∂⁡Ωδ. Now, we consider a decreasing sequence of δ’s, {δn}n≥1, with δ1<δ0, such that limn→∞⁡δn=0. Then, for every n≥1,

D δ n ⊂ D δ n + 1 ⊂ Ω ∖ Ω ¯ 0 , Ω ∖ Ω ¯ 0 = ⋃ m ≥ n D δ m .

Moreover, for every n≥1, ε∈(0,ε0⁢(δn)) and η∈(0,η0⁢(δn)), it follows from (4.13) that

(4.14) θ ε ≤ L a - η , D δ n min in ⁢ D δ n ≡ Ω ∖ Ω ¯ δ n .

Pick a sequence {εm}m≥1 of ε’s such that limm→∞⁡εm=0, as well as an integer n≥1. Then, thanks to (4.14), for sufficiently large m≥1, say m≥m0⁢(n), we have that θεm≤La-η0⁢(δn+2)/2,Dδn+2min in Dδn+2. In particular, there exists a constant Cn+1>0 such that

θ ε m ≤ L a - η 0 ⁢ ( δ n + 2 ) / 2 , D δ n + 2 min ≤ C n + 1 in ⁢ D δ n + 1 .

Thus, by the global Schauder estimates, there exists a constant C~n such that

∥ θ ε m ∥ 𝒞 2 + μ ⁢ ( D ¯ δ n ) ≤ C ~ n for all ⁢ m ≥ m 0 ⁢ ( n ) .

Consequently, as the injection 𝒞2+μ⁢(D¯δn)↪𝒞2⁢(D¯δn) is compact, there exists a subsequence of θεm, denoted by {θm,n}m≥1, and a function Θn∈𝒞2⁢(D¯δn) such that

lim m → ∞ ⁡ θ m , n = Θ n in ⁢ 𝒞 2 ⁢ ( D ¯ δ n ) .

As this scheme can be repeated for every integer n≥1, we can extract a subsequence of θεm, denoted by {θm,1}m≥1, such that

lim m → ∞ ⁡ θ m , 1 = Θ 1 in ⁢ 𝒞 2 ⁢ ( D ¯ δ 1 ) ,

and a subsequence of {θm,1}m≥1, denoted by {θm,2}m≥1, such that

lim m → ∞ ⁡ θ m , 2 = Θ 2 in ⁢ 𝒞 2 ⁢ ( D ¯ δ 2 ) ,

and, more generally, for every n≥2, a subsequence of {θm,n-1}m≥1, denoted by {θm,n}m≥1, such that

lim m → ∞ ⁡ θ m , n = Θ n in ⁢ 𝒞 2 ⁢ ( D ¯ δ n ) ,

where Θn∈𝒞2⁢(D¯δn) for all n≥1. By the uniqueness of the limits,

Θ n + 1 | D ¯ δ n = Θ n for all ⁢ n ≥ 1 .

So Θn+1 provides us with an extension of Θn to D¯δn+1 for all n≥1. Moreover, by (4.1), letting m→∞ in the differential equation of θm,n, it becomes apparent that, for every n≥1, the function Θn solves the problem

{ - Δ ⁢ u = λ ⁢ m ⁢ ( x ) ⁢ u - a ⁢ u p ⁢ ( x ) in ⁢ D δ n , B ⁢ u = 0 on ⁢ ∂ ⁡ Ω .

Furthermore, since limn→∞⁡δn=0, it follows from (4.10) that the diagonal subsequence, {θn,n}n≥1, converges to a large solution of the singular problem (1.12). This ends the proof of (1.11) for α=0.

Step 3: Proof of (1.9)–(1.11) for β>α>0. Suppose β>α>0. In this case, the function ωε≡εαp-1⁢θε solves the problem

{ - Δ ⁢ w = λ ⁢ m ⁢ ( x ) ⁢ w - ( a ε ε α + b ε ε α ) ⁢ w p in ⁢ Ω , B ⁢ w = 0 on ⁢ ∂ ⁡ Ω .

Since β>α, (Ha) and (Hb) imply that

lim ε ↓ 0 ⁡ ( a ε ε α + b ε ε α ) = a + lim ε ↓ 0 ⁡ ( ε β - α ⁢ b ε ε β ) = a uniformly in ⁢ Ω ¯ .

Therefore, (1.9), (1.10) and (1.11) follow easily by applying Steps 1 and 2 to ωε.

Step 4: Proof of (1.14) for β>α≥0. Set ϑε≡εβ-αp-1⁢ωε=εβp-1⁢θε. Then ϑε satisfies

(4.15) { - Δ ⁢ ϑ ε = λ ⁢ m ⁢ ϑ ε - ( a ε ε β + b ε ε β ) ⁢ ϑ ε p in ⁢ Ω , B ⁢ ϑ ε = 0 on ⁢ ∂ ⁡ Ω .

According to (Ha), (Hb) and (1.13), we have that

lim ε ↓ 0 ⁡ ( a ε ε β + b ε ε β ) = lim ε ↓ 0 ⁡ ( ε α - β ⁢ a ε ε α ) + b = { + ∞ in ⁢ Ω ∖ Ω ¯ 0 , b in ⁢ Ω ¯ 0 .

Set η:=12minΩ¯b∈(0,minΩ¯b), and let ε0∈(0,1) be such that

b ε ε β ≥ b - η for all ⁢ ε ∈ ( 0 , ε 0 ) .

Then

c ε ≡ a ε ε β + b ε ε β ≥ b - η for all ⁢ ε ∈ ( 0 , ε 0 ) ,

and it follows from (4.15) that

- Δ ⁢ ϑ ε = λ ⁢ m ⁢ ϑ ε - c ε ⁢ ϑ ε p ≤ λ ⁢ m ⁢ ϑ ε - ( b - η ) ⁢ ϑ ε p .

Thus, by Lemma 2.1, it is apparent that

(4.16) ϑ ε ≤ U b - η in ⁢ Ω for all ⁢ ε ∈ ( 0 , ε 0 ) .

In particular, {ϑε}ε∈(0,ε0) is bounded in L∞⁢(Ω). Moreover, multiplying the differential equation of (4.15) by ϑε and integrating by parts in Ω yields to

(4.17) ∫ Ω | ∇ ⁡ ϑ ε | 2 ⁢ 𝑑 x + ∫ Γ 1 γ ⁢ ϑ ε 2 ⁢ 𝑑 S + ∫ Ω c ε ⁢ ϑ ε p + 1 ⁢ 𝑑 x = λ ⁢ ∫ Ω m ⁢ ϑ ε 2 ⁢ 𝑑 x .

Thus, since γ≥0 on Γ1 and cε≥0 in Ω, it follows from (4.17) that {ϑε}ε∈(0,ε0) is bounded in W1,2⁢(Ω). As the injection W1,2⁢(Ω)↪L2⁢(Ω) is compact, for any given sequence εn∈(0,ε0), n≥1, such that limn→∞⁡εn=0, one can extract a subsequence, relabeled by n≥1, such that, for some ϑ0∈L2⁢(Ω), limn→∞⁡ϑεn=ϑ0 in L2⁢(Ω). Moreover, by (4.16) and (4.17), we also find that

∫ Ω ∖ Ω ¯ 0 a ε n ε n β ⁢ ϑ ε n p + 1 ⁢ 𝑑 x ≤ ∫ Ω c ε n ⁢ ϑ ε n p + 1 ⁢ 𝑑 x ≤ λ ⁢ ∫ Ω m ⁢ ϑ ε 2 ⁢ 𝑑 x ≤ λ ⁢ max Ω ¯ ⁡ m ⁢ ∫ Ω U b - η 2 ⁢ 𝑑 x .

Hence, there is a constant C>0 such that

(4.18) ∫ Ω ∖ Ω ¯ 0 a ε n ε n β ⁢ ϑ ε n p + 1 ⁢ 𝑑 x ≤ C for all ⁢ n ≥ 1 .

Consequently, since β>0 and

lim n → ∞ ⁡ a ε n ε n β = + ∞ pointwise in ⁢ Ω ∖ Ω ¯ 0 ,

it follows from (4.18) that

lim n → ∞ ⁡ ϑ ε n = 0 almost everywhere in ⁢ Ω ∖ Ω ¯ 0 .

Thus, by the uniqueness of the limit, ϑ0=0 in Ω∖Ω¯0. To complete the proof, it remains to show that ϑ0=Ub,Ω0 in Ω0, where Ub,Ω0 stands for the unique positive solution of (1.15). This can be proven with the following argument.

Note that, by the definition of ϑεn, we have that

(4.19) c ε n ⁢ ϑ ε n p = c ε n ⁢ ε n ( β - α ) ⁢ p p - 1 ⁢ ω ε n p = c ε n ⁢ ε n β - α + β - α p - 1 ⁢ ω ε n p = a ε n + b ε n ε n α ⁢ ε n β - α p - 1 ⁢ ω ε n p .

Thus, since

lim n → ∞ ⁡ a ε n + b ε n ε n α = a + lim n → ∞ ⁡ ( b ε n ε n β ⁢ ε n β - α ) = a

uniformly in Ω¯ and, owing to (1.11), we already know that limn→∞⁡ωεn=L in Ω¯∖Ω¯0, it becomes apparent that

c ε n ⁢ ϑ ε n p ≈ a ⁢ L p ⁢ ε n β - α p - 1 as ⁢ n → ∞ .

In particular,

lim n → ∞ ⁡ ( c ε n ⁢ ϑ ε n p ) = 0 uniformly in compact subsets of ⁢ ( Ω ∪ ∂ ⁡ Ω ) ∖ Ω ¯ 0 .

This nice convergence is lost at ∂⁡Ω0 because L=∞ on ∂⁡Ω0. This technical difficulty is overcome with the integrability condition Lmin∈L2⁢p⁢(Ω∖∂⁡Ω0) and the assumption that

(4.20) a ε + b ε ≥ ε α ⁢ a for sufficiently small ⁢ ε > 0 .

Next, we will show that {ϑεn}n≥1 is actually a Cauchy sequence in W1,2⁢(Ω). This entails that ϑ0∈W1,2⁢(Ω) and that limn→∞⁡ϑεn=ϑ0 in W1,2⁢(Ω). Actually, since ϑ0=0 in Ω∖Ω¯0, it becomes apparent that ϑ0∈W01,2⁢(Ω∖Ω¯δ) for sufficiently small δ, say δ<δ0, where we are denoting Ωδ≡{x∈Ω:dist⁡(x,Ω0)<δ}. Thus, since Ω0 is stable, as discussed, e.g., in [8], we find that

ϑ 0 ∈ ⋂ δ ∈ ( 0 , δ 0 ) W 0 1 , 2 ⁢ ( Ω ∖ Ω ¯ δ ) = W 0 1 , 2 ⁢ ( Ω ∖ Ω ¯ 0 ) .

To prove that {ϑεn}n≥1 is a Cauchy sequence in W1,2⁢(Ω), we argue as usual. By some direct straightforward manipulations from (4.15), it is easily seen that, for every pair of integers, n≥1 and ℓ≥1,

Thus, since γ≥0 on Γ1, we find that

To estimate the right-hand side of (4.21), we proceed as follows. By (4.20), there exists ε1∈(0,ε0) such that, for every ε∈(0,ε1), (4.20) implies that

- Δ ⁢ ω ε = λ ⁢ m ⁢ ω ε - ( a ε ε α + b ε ε α ) ⁢ ω ε p ≤ λ ⁢ m ⁢ ω ε - a ⁢ ω ε p

for all ε∈(0,ε1). Thus, by Lemma 2.1, it becomes apparent that

(4.22) ω ε ≤ L min in ⁢ Ω ∖ Ω ¯ 0

for all ε∈(0,ε1). In particular, by (1.11), we find that L=Lmin because Lmin≤L by definition of minimal solution. By the Hölder inequality,

(4.23) ∫ Ω c ε n ⁢ ϑ ε n p ⁢ | ϑ ε n - ϑ ε ℓ | ⁢ 𝑑 x ≤ ( ∫ Ω c ε n 2 ⁢ ϑ ε n 2 ⁢ p ⁢ 𝑑 x ) 1 2 ⁢ ∥ ϑ ε n - ϑ ε ℓ ∥ L 2 ⁢ ( Ω ) .

Moreover,

∫ Ω c ε n 2 ⁢ ϑ ε n 2 ⁢ p ⁢ 𝑑 x = ∫ Ω ∖ Ω ¯ 0 c ε n 2 ⁢ ϑ ε n 2 ⁢ p ⁢ 𝑑 x + ∫ Ω 0 c ε n 2 ⁢ ϑ ε n 2 ⁢ p ⁢ 𝑑 x .

According to (4.19) and (4.22), there exists a constant C>0 such that

∫ Ω ∖ Ω ¯ 0 c ε n 2 ⁢ ϑ ε n 2 ⁢ p ⁢ 𝑑 x = ε n 2 ⁢ ( β - α ) p - 1 ⁢ ∫ Ω ∖ Ω ¯ 0 a ε n + b ε n ε n α ⁢ ω ε n 2 ⁢ p ⁢ 𝑑 x ≤ ε n 2 ⁢ ( β - α ) p - 1 ⁢ C ⁢ ∫ Ω ∖ Ω ¯ 0 ( L min ) 2 ⁢ p ⁢ 𝑑 x .

Hence, since Lmin∈L2⁢p⁢(Ω∖∂⁡Ω0), we find that

lim n → ∞ ⁡ ∫ Ω ∖ Ω ¯ 0 c ε n 2 ⁢ ϑ ε n 2 ⁢ p ⁢ 𝑑 x = 0 .

On the other hand, as, due to (1.13),

lim ε ↓ 0 ⁡ c ε = b uniformly in ⁢ Ω ¯ 0 ,

by (4.16), we find that, for sufficiently large n≥1,

∫ Ω 0 c ε n 2 ⁢ ϑ ε n 2 ⁢ p ⁢ 𝑑 x ≤ ∫ Ω 0 ( b + η ) 2 ⁢ U b - η 2 ⁢ p ⁢ 𝑑 x ≤ C

for some constant C>0. Therefore, by (4.23), there exists a constant C~>0 such that

∫ Ω c ε n ⁢ ϑ ε n p ⁢ | ϑ ε n - ϑ ε ℓ | ⁢ 𝑑 x ≤ C ~ ⁢ ∥ ϑ ε n - ϑ ε ℓ ∥ L 2 ⁢ ( Ω ) .

By symmetry,

∫ Ω c ε ℓ ⁢ ϑ ε ℓ p ⁢ | ϑ ε n - ϑ ε ℓ | ⁢ 𝑑 x ≤ C ~ ⁢ ∥ ϑ ε n - ϑ ε ℓ ∥ L 2 ⁢ ( Ω ) .

From these estimates, it readily follows that ϑεn is a Cauchy sequence in W1,2⁢(Ω0). The fact that ϑ0 provides us with a weak non-negative solution of (1.15) is routine from the weak formulation of the ϑεn-problem by letting n→∞. To show that ϑ0≫0, we can use the following argument. Since, for every n≥1, ϑεn⁢(x)>0 for all x∈∂⁡Ω0, it becomes apparent that ϑεn is a positive supersolution of the problem

(4.24) { - Δ ⁢ v = λ ⁢ m ⁢ v - ( a ε n ε n β + b ε n ε n β ) ⁢ v p in ⁢ Ω 0 , v = 0 on ⁢ ∂ ⁡ Ω 0 .

Thus, by Lemma 2.1, we have that, for every n≥1,

(4.25) ϑ ε n ≥ v ε n in ⁢ Ω 0 ,

where vεn denotes the unique positive solution of (4.24). On the other hand, by our assumptions of the weight functions aε and bε and, in particular, by (1.13) and (Hb), we have that

lim ε ↓ 0 ⁡ ( a ε ε β + b ε ε β ) = b uniformly in ⁢ Ω ¯ 0 .

Consequently, according to Theorem 1.1, it is apparent that

lim n → ∞ ⁡ v ε n = U b , Ω 0 ,

where Ub,Ω0 stands for the (unique) positive solution of (1.15). Therefore, letting n→∞ in (4.25) yields to ϑ0≥Ub,Ω0. Actually, by the uniqueness of the weak positive solution, we have that ϑ0=Ub,Ω0. As this argument can be repeated along any sequence {εn}n≥1, the proof of Theorem 1.3 is completed.

Communicated by Guozhen Lu

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11731005

Award Identifier / Grant number: 11401277

Funding source: Ministerio de Ciencia, Innovación y Universidades

Award Identifier / Grant number: PGC2018-097104-B-I00

Funding source: Fundamental Research Funds for the Central Universities

Award Identifier / Grant number: lzujbky-2021-52

Funding statement: W. T. Li was partially supported by NSF of China (11731005), J. López-Gómez was partially supported by the IMI of Complutense University, and the Ministry of Science, Innovation and Universities of Spain under Grant PGC2018-097104-B-I00, and J. W. Sun was partially supported by the NSF of China (11401277) and the Fundamental Research Funds for the Central Universities (lzujbky-2021-52).

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Received: 2021-09-10

Revised: 2021-10-01

Accepted: 2021-10-02

Published Online: 2021-10-15

Published in Print: 2021-11-01

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

Sharp Blow-Up Profiles of Positive Solutions for a Class of Semilinear Elliptic Problems

Abstract

1 Introduction and Main Results

Theorem 1.1.

Theorem 1.2.

Theorem 1.3.

2 A Useful Comparison Result

Lemma 2.1.

Proof.

3 Proof of Theorems 1.1 and 1.2

Theorem 3.1.

Proof.

3.1 Proof of Theorem 1.1

3.2 Proof of Theorem 1.2

4 Proof of Theorem 1.3

References

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