On the Existence of Solutions of a Nonlocal Elliptic Equation with a 𝑝-Kirchhoff-Type Term

Abstract

Questions on the existence of positive solutions for the following class of elliptic problems are studied: −[𝑀(‖𝑢‖^𝑝_1,𝑝)]^1,𝑝Δ_𝑝𝑢=𝑓(𝑥,𝑢), in Ω, 𝑢=0, on 𝜕Ω, where Ω⊂ℝ^𝑁 is a bounded smooth domain, 𝑓∶Ω×ℝ⁺→ℝ and 𝑀∶ℝ⁺→ℝ,ℝ⁺=[0,∞) are given functions.

1. Introduction

In this paper, we are concerned with the elliptic problem−[𝑀(‖𝑢‖^𝑝_1,𝑝)]^1,𝑝Δ_𝑝𝑢=𝑓(𝑥,𝑢)inΩ,𝑢=0on𝜕Ω, (1.1)where Ω⊂𝑅^𝑁 is a bounded smooth domain, 𝑓∶Ω×ℝ⁺→ℝ and 𝑀∶ℝ⁺→ℝ,ℝ⁺=[0,∞) are given functions, Δ_𝑝 is the 𝑝-Laplacian: Δ_𝑝𝑢=div(|∇𝑢|^𝑝−2∇𝑢),𝑝>1, (1.2)and ‖⋅‖_1,𝑝 is the usual norm‖𝑢‖^𝑝_1,𝑝=_Ω|∇𝑢|^𝑝 (1.3)in the Sobolev space 𝑊₀^1,𝑝(Ω).

Such a problem, which will be named 𝑝-Kirchhoff problem, is a generalization of the classical stationary Kirchhoff equation−[𝑀(‖𝑢‖²)]Δ𝑢=𝑓(𝑥,𝑢)inΩ,𝑢=0on𝜕Ω, (1.4)where ‖𝑢‖²=∫_Ω|∇𝑢|² is the usual norm in 𝐻¹₀(Ω).

As it is well known, problem (1.4) is the stationary counterpart of the hyperbolic Kirchhoff equation𝜌𝜕²𝑢𝜕𝑡²𝑃−(⁰ℎ+𝐸2𝐿^𝐿₀|𝜕𝑢|𝜕𝑥²𝜕𝑑𝑥)²𝑢𝜕𝑥²=0 (1.5)that appeared at the first time in the work of Kirchhoff [1], in 1883. The equation in (1.5) is called Kirchhoff equation and it extends the classical D'Alembert wave equation, by considering the effects of the changes in the length of the strings during the vibrations.

The interest of the mathematicians on the so-called nonlocal problems like (1.1) (nonlocal because of the presence of the term 𝑀(‖𝑢‖^𝑝_1,𝑝), which implies that equations in (1.1) and (1.4) are no longer pointwise equalities) has increased because they represent a variety of relevant physical and engineering situations and requires a nontrivial apparatus to solve them.

Particularly, problem (1.1) presents some combinations that, at least to our knowledge, seem to be new. Indeed, in problem (1.1) appears the nonlocal term 𝑀(‖𝑢‖^𝑝_1,𝑝) motivated, among other things, by the above physical situations. Furthermore, we have the presence of the 𝑝-Laplacian operator that appears in several areas of the science such as astronomy, glaciology, climatology, nonnewtonian fluids, petroleum extraction. Problems that involve these two terms, 𝑀(‖𝑢‖^𝑝_1,𝑝) and Δ_𝑝𝑢, present several difficulties such as uniqueness, regularity, degeneracy, as we will see throughout this paper.

Beside these considerations, we also consider a case with the presence of a singular term which poses an additional difficulty in our study. Singular elliptic problems arise in chemical heterogeneous catalysts, nonnewtonian fluids, nonlinear heat conduction, among other phenomena.

In case 𝑝=2, problem (1.1) has been studied by several authors. See [2–7], and the references therein. Particularly, this work was motivated by [2–4, 6].

We will establish existence results for problem (1.1) by considering several classes of functions 𝑀 and 𝑓.

An outline of this work is as follows:

In Section 2, we recall some properties of the 𝑝-Laplacian. In Section 3, we study the case in which 𝑓 depends only on 𝑥∈Ω. This is the 𝑀_𝑝 -Linear case. In Section 4, we attack problem (1.1) when 𝑓 is sublinear, that is, 𝑓(𝑢)=𝑢^𝛼, for some 0<𝛼<1.

In both Sections 3 and 4, we suitably adapt ideas developed in [2, 3, 6].

In Section 5, we analyze the case in which 𝑓 possesses a singular term. More precisely, 𝑓 is of the following form:𝑓(𝑥,𝑢)=ℎ(𝑥)𝑢^{𝛾−𝑝+2}+𝑢^{𝛼−𝑝+2}, (1.6)for 𝑥∈Ω and 𝑢>0, with 𝛾,𝛼∈(𝑝−2,𝑝−1). In this section, we have to use some arguments different from those in [4].

To end this introduction we recall that 𝑢∈𝑊₀^1,𝑝(Ω) is a weak solution of problem (1.1) if [𝑀(‖𝑢‖^𝑝_1,𝑝)]^𝑝−1_Ω|∇𝑢|^𝑝−2∇𝑢∇𝜙=_Ω𝑓(𝑥,𝑢)𝜙,∀𝜙∈𝑊₀^1,𝑝(Ω). (1.7)

2. Preliminaries on The 𝑝-Laplacian

We will briefly expose some properties of the 𝑝-Laplacian operator defined byΔ_𝑝𝑢≡div(∇𝑢|^𝑝−2∇𝑢),1<𝑝<∞. (2.1)

First, we consider the problem−Δ_𝑝𝑢=𝑓(𝑥),inΩ,𝑢=0,on𝜕Ω, (2.2)where 𝑓∈𝑊^−1,𝑝′(Ω),𝑝^′=𝑝/(𝑝−1),𝑝>1, and the boundary condition will be understood as 𝑢∈𝑊₀^1,𝑝(Ω).

The following results holds as a simple consequence of a minimization of a suitable functional.

Theorem 2.1. If 𝑓∈𝑊^−1,𝑝′(Ω), then problem (2.2) has only a solution 𝑢∈𝑊₀^1,𝑝(Ω) in the weak sense, namely,[𝑀(‖𝑢‖^𝑝_1,𝑝)]^𝑝−1_Ω|∇𝑢|^𝑝−2∇𝑢∇𝜙=_Ω𝑓(𝑥)𝜙,∀𝜙∈𝑊₀^1,𝑝(Ω). (2.3)

So, we have defined an operator (−Δ_𝑝)⁻¹∶𝑊^−1,𝑝′(Ω)→𝑊₀^1,𝑝(Ω), the inverse of −Δ_𝑝, which satisfies the following.

(a) −Δ_𝑝∶𝑊₀^1,𝑝(Ω)→𝑊^−1,𝑝′(Ω) is uniformly continuous on bounded sets, where such operator is defined as−Δ_𝑝∶𝑊₀^1,𝑝(Ω)→𝑊^−1,𝑝′(Ω),𝑢↦−Δ_𝑝𝑢, (2.4)where⟨−Δ_𝑝𝑢,𝜙⟩=_Ω|∇𝑢|^𝑝−2∇𝑢∇𝜙,∀𝜙∈𝑊₀^1,𝑝(Ω). (2.5)Here, we denote by ⟨,⟩ the duality pairing between 𝑊^−1,𝑝′(Ω) and 𝑊₀^1,𝑝(Ω).

(b) If 𝑓∈𝐶⁰(Ω), then the weak solution of (2.2) belongs to 𝐶^1,𝛼(Ω), for some 0<𝛼<1, and the mapping (−Δ_𝑝)⁻¹∶𝐶⁰(Ω)→𝐶¹(Ω) is compact.

Theorem 2.2 (weak comparison principle). Let 𝑢₁,𝑢₂∈𝑊₀^1,𝑝(Ω) satisfy−Δ_𝑝𝑢₁≤−Δ_𝑝𝑢₂,inΩ,(intheweaksense)𝑢₁≤𝑢₂,on𝜕Ω. (2.6)Then, 𝑢₁≤𝑢₂ a.e. in Ω.

Theorem 2.3 (a Hopf-type maximum principle). If 𝑢∈𝐶¹(Ω)∩𝑊₀^1,𝑝(Ω) and verifies−Δ_𝑝𝑢≥0inΩ,𝑢>0inΩ,𝑢=0on𝜕Ω, (2.7)then 𝜕𝑢/𝜕𝜂<0 on 𝜕Ω, where 𝜂 is the outward normal to 𝜕Ω.

Theorem 2.4 (a strong maximum principle). Assume that 𝑘∈ℝ is a nonnegative number 1<𝑝≤2 and Ω is a bounded domain ℝ^𝑁. Suppose that 𝑢∈𝐶¹(Ω) satisfies−Δ_𝑝𝑢+𝑘𝑢≥0inΩ(intheweaksense), (2.8)𝑢≥0 and 𝑢≢0 in Ω. Then, 𝑢>0 in Ω. The conclusion is still true for all 𝑝>1 when 𝑘=0.

We now consider the eigenvalue problem−Δ_𝑝𝑢=𝜆|𝑢|^𝑝−2𝑢inΩ,𝑢=0on𝜕Ω, (2.9)where 𝜆∈ℝ is a parameter. We say that 𝜆∈ℝ is an eigenvalue of (2.9) if there exists a function 𝑢∈𝑊₀^1,𝑝(Ω),𝑢≢0, satisfying (2.9) in the weak sense. Such a function is called an eigenfunction of (2.9) associated to the eigenvalue 𝜆.

There exists the first positive eigenvalue 𝜆₁ of problem (2.9) which is characterized as the minimum of the Rayleigh quotient𝜆₁=min_{0≠𝑢∈𝑊01,𝑝(Ω)}∫_Ω|∇𝑢|^𝑝∫_Ω|𝑢|^𝑝>0. (2.10)Moreover, 𝜆₁ is simple (i.e., all the associated first eigenfunctions 𝑢 are merely constant multiples of each other) and isolated (i.e., there are no eigenvalues less than 𝜆₁ and no eigenvalues in some right reduced neighborhood of 𝜆₁). There is a positive (in Ω) eigenfunction 𝜑₁ corresponding to 𝜆₁.

For more informations on the 𝑝-Laplacian the reader may consult [8–11].

3. The 𝑀_𝑝-Linear Case

This section is devoted to the study of the so-called 𝑀_𝑝-Linear problem−[𝑀(‖𝑢‖^𝑝_1,𝑝)]^𝑝−1Δ_𝑝𝑢=𝑓inΩ,𝑢=0on𝜕Ω, (3.1)where 𝑝>1,𝑓∈𝑊^−1,𝑝′(Ω),𝑝^′ is the conjugate exponent of 𝑝, that is, 𝑝^′=𝑝/(𝑝−1) and 𝑊^−1,𝑝′(Ω) is the topological dual of 𝑊₀^1,𝑝(Ω).

The next result is an adaptation of some ideas contained in [12, 13] (which were used for another nonlocal problem) for problem (3.1).

Theorem 3.1. For each 0≠𝑓∈𝑊^−1,𝑝′(Ω), problem (3.1) possesses as many solutions as the folloing equation:𝑀(𝑡)𝑡^1/𝑝=‖𝜔‖_1,𝑝,𝑡>0, (3.2) where 𝜔∈𝑊₀^1,𝑝(Ω) is the only solution of problem (2.2).

Proof. First, let us suppose that 𝑢∈𝑊₀^1,𝑝(Ω) is a solution of (3.1), where 0≠𝑓∈𝑊^−1,𝑝′(Ω) is fixed. Hencediv(|∇(𝑀(‖𝑢‖^𝑝_1,𝑝)𝑢)|^𝑝−2∇(𝑀(‖𝑢‖^𝑝_1,𝑝)𝑢))=𝑓 (3.3)and so 𝜔=𝑀(‖𝑢‖^𝑝_1,𝑝)𝑢 is the solution of−Δ_𝑝𝜔=𝑓inΩ,𝜔=0on𝜕Ω, (3.4)and observe that ‖𝜔‖_1,𝑝=𝑀(‖𝑢‖^𝑝_1,𝑝)‖𝑢‖_1,𝑝 from which we conclude that 𝑡=‖𝑢‖^𝑝_1,𝑝 is a solution of (3.2).
Conversely, let 𝜔 be the solution of (3.4) and suppose that 𝑡>0 is a solution of (3.2).
Define𝑢=𝑡^1/𝑝𝜔‖𝜔‖_1,𝑝, (3.5)and so ‖𝑢‖_1,𝑝=𝑡^1/𝑝. A straightforward calculation shows that−[𝑀(‖𝑢‖^𝑝_1,𝑝)]^𝑝−1Δ_𝑝𝑢=−[𝑀(𝑡)𝑡^1/𝑝‖𝜔‖_1,𝑝]^𝑝−1Δ_𝑝𝜔inΩ, (3.6)and, because 𝑡>0 is a solution of 𝑀(𝑡)𝑡^1/𝑝=‖𝜔‖_1,𝑝, one has (𝑀(𝑡)𝑡^1/𝑝/‖𝜔‖_1,𝑝)=1. Hence,−[𝑀(‖𝑢‖^𝑝_1,𝑝)]^𝑝−1Δ_𝑝𝑢=𝑓inΩ,𝑢=0on𝜕Ω, (3.7)which concludes the proof of the theorem.

Remark 3.2. Let us suppose that 𝑀∶ℝ⁺→ℝ⁺ is a continuous function. So,lim_𝑡→0+𝑀(𝑡)𝑡^1/𝑝=0. (3.8)

Assume, in addition, that there are 𝑡₀,𝑚₀>0 such that 𝑀(𝑡)≥𝑚₀>0 for 𝑡≥𝑡₀. In this case,lim_𝑡→+∞𝑀(𝑡)𝑡^1/𝑝=+∞. (3.9)Because of the intermediate value theorem, for ‖𝜔‖_1,𝑝, there is 𝑡>0 such that𝑀(𝑡)𝑡^1/𝑝=‖𝜔‖_1,𝑝. (3.10)In particular, if 𝑀(𝑡)𝑡^1/𝑝 is an increasing function for 𝑡>0, problem (3.1) possesses only a solution for each 0≠𝑓∈𝑊^−1,𝑝′(Ω). As an example, we take 𝑀(𝑡)=𝑒^−𝑡 for 𝑡≥0. Thus, problem (3.1) possesses as many solutions as the following equation:𝑒^−𝑡𝑡^1/𝑝=‖𝜔‖_1,𝑝. (3.11)

A simple exercise shows that the function 𝑔(𝑡)=𝑒^−𝑡𝑡^1/𝑝 attains its maximum in 1/𝑝 and 𝑔(1/𝑝)=1/𝑒√^1/𝑝𝑝𝑝. Consequently, if ‖𝜔‖_1,𝑝>1/𝑒√^1/𝑝𝑝𝑝, problem (3.1) does not possess any solution; if ‖𝜔‖_1,𝑝=1/𝑒√^1/𝑝𝑝𝑝, problem (3.1) possesses exactly one solution; and if 0<‖𝜔‖_1,𝑝<1/𝑒√^1/𝑝𝑝𝑝, problem (3.1) possesses exactly two solutions.

As we see, the presence of the term [𝑀(‖𝑢‖^𝑝_1,𝑝)]^𝑝−1 produces great difference between problems (2.2) and (3.1).

Remark 3.3. Let us consider the case 𝑓=0, that is,−[𝑀(‖𝑢‖^𝑝_1,𝑝)]^𝑝−1Δ_𝑝𝑢=0inΩ,𝑢=0on𝜕Ω. (3.12)

If 𝑀(𝑡)>0 for all 𝑡≥0, problem (3.12) possesses only the null solution. If 𝑀(𝑡₀)=0, for some 𝑡₀>0, problem (3.12) possesses infinitely many solutions. Indeed, if 𝑢∈𝑊₀^1,𝑝(Ω),𝑢≠0, we have that𝑣=𝑡₀^1/𝑝𝑢‖𝑢‖_1,𝑝 (3.13)is a solution of (3.12).

4. on A Sublinear Problem

In this section, we are going to study the sublinear problem−[𝑀(‖𝑢‖^𝑝_1,𝑝)]^𝑝−1Δ_𝑝𝑢=𝑢^𝛼inΩ,𝑢>0inΩ,𝑢=0on𝜕Ω, (4.1)where 𝛼 satisfies 0<𝛼<𝑝−1 with 𝑝>1.

Such a kind of problem belongs to a class of problems known as sublinear whose prototype isΔ𝑢=𝑢^𝛼inΩ,𝑢>0inΩ,𝑢=0on𝜕Ω, (4.2)with 0<𝛼<1 (note that, in this case, 𝑝=2), which has been vastly studied. See [14–17]. For the nonlocal problem, with 𝑝=2, we cite [2, 3, 6, 7], and the references therein.

In particular, Díaz and Saá [16] study the problemΔ_𝑝𝑢=𝑢^𝛼inΩ,𝑢>0inΩ,𝑢=0on𝜕Ω, (4.3)1<𝑝<∞,0<𝛼<𝑝−1, and show that it possesses only a positive solution 𝑢∈𝑊₀^1,𝑝(Ω)∩𝐿^∞(Ω). Indeed, such a solution also belongs to 𝑊^2,2_loc(Ω). Our next result describes what happens with the nonlocal sublinear problem (4.1).

Theorem 4.1. Suppose that 𝑀∶ℝ⁺→ℝ is a continuous function satisfying 𝑀(𝑡)>0 for all 𝑡≥0. Then problem (4.1) has at least as many solutions as the equation[𝑀(𝑡)]^𝑝−1⋅𝑡^{(𝑝−1−𝛼)/𝑝}=‖𝑣‖^{𝑝−1−𝛼}_1,𝑝,𝑡>0, (4.4)where 𝑣 is the solution of (4.1).

Proof. We adapt for problem (4.1) the ideas developed in [3, 6]. Let us suppose that 𝑡>0 is a solution of the (4.4) and set 𝛾=𝑡^1/𝑝/‖𝑣‖_1,𝑝 which implies ‖𝛾𝑣‖_1,𝑝=𝑡^1/𝑝. So, a simple calculation shows that[𝑀(‖𝛾𝑣‖^𝑝_1,𝑝)]^𝑝−1=𝛾^{𝛼−(𝑝−1)}. (4.5)
Define 𝑢=𝛾𝑣 and let us show that such a function 𝑢 is a solution of problem (4.1). Indeed, this follows from the calculation below:−[𝑀(‖𝑢‖^𝑝_1,𝑝)]^𝑝−1Δ_𝑝𝑢=−[𝑀(‖𝛾𝑣‖^𝑝_1,𝑝)]^𝑝−1Δ_𝑝(𝛾𝑣)=−[𝑀(‖𝛾𝑣‖^𝑝_1,𝑝)]^𝑝−1div(|∇(𝛾𝑣)|^𝑝−2∇(𝛾𝑣))=−[𝑀(‖𝛾𝑣‖^𝑝_1,𝑝)]^𝑝−1div(𝛾^𝑝−2𝛾|∇𝑣|^𝑝−2∇𝑣)=[𝑀(‖𝛾𝑣‖^𝑝_1,𝑝)]^𝑝−1𝛾^𝑝−1(−Δ_𝑝𝑣)=𝛾^{𝛼−(𝑝−1)}𝛾^𝑝−1(−Δ_𝑝𝑣)=𝛾^𝛼𝑣^𝛼=(𝛾𝑣)^𝛼=𝑢^𝛼inΩ. (4.6)Hence, 𝑢 is a solution of problem (4.1).

Remark 4.2. Remark 3.2, mutatis mutandis, remains valid for problem (4.1).

5. A Singular Problem Via The Galerkin Method

In this section, we will study problem−[𝑀(‖𝑢‖^𝑝_1,𝑝)]^𝑝−1Δ_𝑝𝑢=ℎ(𝑥)𝑢^{𝛾−𝑝+2}+𝑢^{𝛼−𝑝+2}inΩ,𝑢>0inΩ,𝑢=0on𝜕Ω, (5.1)which is a singular perturbation of problem (4.1), where Ω⊂ℝ^𝑁 is a bounded smooth domain, 2≤𝑝<𝑁 and ℎ is a suitable function defined in Ω.

We will attack problem (5.1) by using the Galerkin method which rests heavily on the following result, which is a variant of the well-known Brouwer fixed point theorem, whose proof may be found in Lions [18].

Proposition 5.1. Suppose 𝐹∶ℝ^𝑚→ℝ^𝑚 is a continuous function such that ⟨𝐹(𝜉),𝜉⟩≥0 with |𝜉|=𝑟, for some 𝑟>0, where ⟨⋅,⋅⟩ denotes the usual norm in ℝ^𝑚. Then, there is 𝜉₀∈𝐵_𝑟(0) such that 𝐹(𝜉₀)=0.

We will suppose that 𝑀∶ℝ⁺→ℝ⁺ satisfies

(𝑀₁) there are 𝑚₀>0 and 𝜃₁>0 such that𝑀(𝑡)≥𝑚₀ if 𝑡≥𝜃₁;(𝑀₂)𝜃₂=sup{𝑡>0;𝑀(𝑡)=0}>0.

If 𝑀(𝑡)≥𝑚₀>0 for all 𝑡≥0, condition (𝑀₂) is vacuous.

Theorem 5.2. Let ℎ∶Ω→ℝ be a continuous function with ℎ>0 on Ω,𝛾,𝛼∈(𝑝−2,𝑝−1),2≤𝑝<𝑁 and 𝑀∶ℝ⁺→ℝ⁺ is a continuous function satisfying (𝑀₁)-(𝑀₂). Then problem (5.1) possesses a positive solution.

We will split the proof of this theorem in several lemmas. First, for each fixed 𝜖>0, we will consider the following problem:−[𝑀(‖𝑢‖^𝑝_1,𝑝)]^𝑝−1Δ_𝑝𝑢=ℎ(𝑥)(𝜖+𝑢)^{𝛾−𝑝+2}+𝑢^{𝛼−𝑝+2},inΩ,𝑢>0,inΩ,𝑢=0,on𝜕Ω. (5.2)

In what follows, throughout Section 5, we are always supposing that 𝑀,ℎ,𝛼, and 𝛾 enjoy assumptions of Theorem 5.2

Lemma 5.3. For each fixed 𝜖>0, problem (5.2) possesses a solution 𝑢_𝜖.

Proof. Let us consider the problem−[𝑀(‖𝑢‖^𝑝_1,𝑝)]^𝑝−1Δ_𝑝𝑢=ℎ(𝑥)(𝜖+|𝑢|)^{𝛾−𝑝+2}+|𝑢|^{𝛼−𝑝+2},inΩ,𝑢=0,on𝜕Ω, (5.3)where 𝑀⁺∶ℝ⁺→ℝ is given by𝑀⁺(𝑡)=0,if0≤𝑡≤𝜃₂,𝑀(𝑡),if𝑡>𝜃₂. (5.4)
We now consider 𝔹={𝑒₁,𝑒₂,𝑒₃,…} a Schauder's basis for 𝑊₀^1,𝑝(Ω) (we recall that a Schauder's basis for a Banach space 𝑋 is a sequence (𝑒_𝑛) such that to each 𝑥∈𝑋 there exists a unique sequence of scalars (𝛼_𝑛) for which the partial sums of ∑𝛼_𝑛𝑒_𝑛 converge to 𝑥 in the norm of 𝑋). For more informations on Schauder's basis see [19] and the references therein.
For each 𝑚∈ℕ, let 𝔹_𝑚=span{𝑒₁,𝑒₂,…} be the finite dimensional vector space spanned by the functions 𝑒₁,…,𝑒_𝑚. So, each 𝑢∈𝔹_𝑚 is written as 𝑢=∑^𝑚_𝑗=1𝜉_𝑗𝑒_𝑗. We will use on 𝔹_𝑚 the norm‖𝑢‖_𝑚=^𝑚_𝑗=1|𝜉_𝑗|. (5.5)
We note that (𝔹_𝑚,‖⋅‖_𝑚) and (ℝ^𝑚,|⋅|_𝑠) are isomorphic through the following map:𝑇∶(𝔹,‖⋅‖_𝑚)→(ℝ,|⋅|_𝑠),𝑢=∑^𝑚_𝑗=1𝜉_𝑗𝑒_𝑗↦𝑇(𝑢)=𝜉=(𝜉₁,𝜉₂,…,𝜉_𝑚), (5.6)with ‖𝑢‖_𝑚=|𝜉|_𝑠=|𝑇(𝑢)|_𝑠 where |𝜉|_𝑠=∑^𝑚_𝑗=1|𝜉_𝑗|.
Since, for each 𝑚∈ℕ,𝔹_𝑚 is a finite dimensional vector space the norms ‖⋅‖_𝑚 and ‖⋅‖_1,𝑝, induced from 𝑊₀^1,𝑝(Ω) on 𝔹_𝑚, are equivalent to each other and so, there exist positive constants 𝑐(𝑚) and 𝑘(𝑚) such that𝑐(𝑚)‖𝑢‖_𝑚≤‖𝑢‖_1,𝑝≤𝑘(𝑚)‖𝑢‖_𝑚. (5.7)
We now consider, for each 𝑚∈ℕ, the following application𝐹∶ℝ^𝑚→ℝ^𝑚,𝜉↦𝐹(𝜉)=(𝐹₁(𝜉),𝐹₂(𝜉),…,𝐹_𝑚(𝜉)), (5.8)where𝐹_𝑗(𝜉)=[𝑀⁺(‖𝑢‖^𝑝_1,𝑝)]^𝑝−1_Ω|∇𝑢|^𝑝−2∇𝑢∇𝑒_𝑗−_Ωℎ(𝑥)𝑒_𝑗(𝜖+|𝑢|)^{𝛾−𝑝+2}−_Ω|𝑢|^{𝛼−𝑝+2}𝑒_𝑗, (5.9)𝑗=1,…,𝑚.
A simple calculation leads us to⟨𝐹(𝜉),𝜉⟩=[𝑀⁺(‖𝑢‖^𝑝_1,𝑝)]^𝑝−1_Ω|∇𝑢|^𝑝−2∇𝑢∇𝑢−_Ωℎ(𝑥)𝑢(𝜖+|𝑢|)^{𝛾−𝑝+2}−_Ω|𝑢|^{𝛼−𝑝+2}𝑢,=[𝑀⁺(‖𝑢‖^𝑝_1,𝑝)]^𝑝−1‖𝑢‖^𝑝_1,𝑝−_Ωℎ(𝑥)𝑢(𝜖+|𝑢|)^{𝛾−𝑝+2}−_Ω|𝑢|^{𝛼−𝑝+2}𝑢. (5.10)We now note that_Ωℎ(𝑥)𝑢(𝜖+|𝑢|)^{𝛾−𝑝+2}≤‖ℎ‖_∞Ω|𝑢|𝜖^{𝛾−𝑝+2}≤𝐶_𝜖‖𝑢‖_1,𝑝, (5.11)and because 𝑝−2<𝛼<𝑝−1, we have 1<𝛼−𝑝+3<2, and so_Ω|𝑢|^{𝛼−𝑝+2}𝑢≤_Ω|𝑢|^{𝛼−𝑝+3}≤𝐶‖𝑢‖^{𝛼−𝑝+3}_1,𝑝. (5.12)Hence,⟨𝐹(𝜉),𝜉⟩≥[𝑀(‖𝑢‖^𝑝_1,𝑝)]^𝑝−1‖𝑢‖^𝑝_1,𝑝−𝐶_𝜖‖𝑢‖_1,𝑝−𝐶‖𝑢‖^{𝛼−𝑝+3}_1,𝑝, (5.13)and from (𝑀₁), for ‖𝑢‖^𝑝_1,𝑝≥𝜃₁, we have 𝑀⁺(‖𝑢‖^𝑝_1,𝑝)=𝑀(‖𝑢‖^𝑝_1,𝑝)≥𝑚₀>0. Thus,⟨𝐹(𝜉),𝜉⟩≥𝑚₀^𝑝−1‖𝑢‖^𝑝_1,𝑝−𝐶_𝜖‖𝑢‖_1,𝑝−𝐶‖𝑢‖^{𝛼−𝑝+3}_1,𝑝≥𝑚₀^𝑝−1[𝑐(𝑚)]^𝑝‖𝑢‖^𝑝_𝑚−𝐶_𝜖𝑘(𝑚)‖𝑢‖_𝑚−[𝑘(𝑚)]^{𝛼−𝑝+3}𝐶‖𝑢‖_𝑚^{𝛼−𝑝+3}≥𝑚₀^𝑝−1𝐶(𝑚)|𝜉|^𝑝_𝑠−𝑁_𝜖(𝑚)|𝜉|_𝑠−𝑄(𝑚)|𝜉|_𝑠^{𝛼−𝑝+3}. (5.14)
We now fix 𝜌_𝑚>0 and for 𝜉∈ℝ^𝑚,|𝜉|_𝑠=𝜌_𝑚 and noting that 1<𝛼−𝑝+3<2≤𝑝, we obtain⟨𝐹(𝜉),𝜉⟩≥𝑚₀^𝑝−1𝐶(𝑚)𝜌^𝑝_𝑚−𝑁_𝜖(𝑚)𝜌_𝑚−𝑄(𝑚)𝜌_𝑚^{𝛼−𝑝+3}. (5.15)So, if 𝜌_𝑚 is large enough, we have⟨𝐹(𝜉),𝜉⟩>0if|𝜉|_𝑠=𝜌_𝑚, (5.16)which implies⟨𝐹(𝜉),𝜉⟩>0if|𝜉|=𝜌^∗_𝑚, (5.17)where 𝜌^∗_𝑚>0, because |⋅| and |⋅|_𝑠 are equivalent in ℝ^𝑚.
From Proposition 5.1 there is 𝜉^𝑚∈ℝ^𝑚, with |𝜉|≤𝜌^∗_𝑚, such that 𝐹(𝜉^𝑚)=0 and note that |𝜉^𝑚|_𝑠≤𝜌_𝑚 for some 𝜌_𝑚 is large enough.
Through the isometric identification of (ℝ^𝑚,|⋅|_𝑠) with (𝔹_𝑚,‖⋅‖_𝑚), we find (𝑢_𝑚)⊂𝔹_𝑚,‖𝑢_𝑚‖_𝑚≤𝜌_𝑚, that is, ‖𝑢_𝑚‖_1,𝑝≤𝜌_𝑚, such that[𝑀⁺(‖𝑢_𝑚‖^𝑝_1,𝑝)]^𝑝−1_Ω|∇𝑢_𝑚|^𝑝−2∇𝑢_𝑚∇𝑒_𝑗−_Ωℎ(𝑥)𝑒_𝑗(𝜖+|𝑢_𝑚|)^{𝛾−𝑝+2}−_Ω|𝑢_𝑚|^{𝛼−𝑝+2}𝑒_𝑗=0, (5.18)for each 𝑗=1,…,𝑚, which gives us[𝑀⁺(‖𝑢_𝑚‖^𝑝_1,𝑝)]^𝑝−1_Ω|∇𝑢_𝑚|^𝑝−2∇𝑢_𝑚∇𝜔−_Ωℎ(𝑥)𝜔(𝜖+|𝑢_𝑚|)^{𝛾−𝑝+2}−_Ω|𝑢_𝑚|^{𝛼−𝑝+2}𝜔=0, (5.19)for all 𝜔∈𝔹_𝑚. Since 𝑢_𝑚∈𝔹_𝑚,[𝑀⁺(‖𝑢_𝑚‖^𝑝_1,𝑝)]^𝑝−1‖𝑢_𝑚‖^𝑝_1,𝑝−_Ωℎ(𝑥)𝑢_𝑚(𝜖+|𝑢_𝑚|)^{𝛾−𝑝+2}−_Ω|𝑢_𝑚|^{𝛼−𝑝+2}𝑢_𝑚=0. (5.20)
Because∫_Ωℎ(𝑥)𝑢_𝑚(𝜖+|𝑢_𝑚|)^{𝛾−𝑝+2}≤‖𝑢‖_∞𝜖^{𝛾−𝑝+2}∫_Ω|𝑢_𝑚|≤𝐶_𝜖‖𝑢_𝑚‖_1,𝑝,∫_Ω|𝑢_𝑚|^{𝛼−𝑝+2}𝑢_𝑚≤𝐶‖𝑢_𝑚‖^{𝛼−𝑝+3}_1,𝑝, (5.21) we obtain[𝑀(‖𝑢_𝑚‖^𝑝_1,𝑝)]^𝑝−1‖𝑢_𝑚‖^𝑝_1,𝑝≤𝐶_𝜖‖𝑢_𝑚‖_1,𝑝+𝐶‖𝑢_𝑚‖^{𝛼−𝑝+3}_1,𝑝. (5.22)
Let us show that the sequence (‖𝑢_𝑚‖_1,𝑝) is bounded. Indeed, suppose, on the contrary, that (‖𝑢_𝑚‖_1,𝑝) is not bounded. So, up to a subsequence, we may suppose that ‖𝑢_𝑚‖_1,𝑝→+∞. From condition (𝑀₁) and inequality (5.22) we obtain0<𝑚₀^𝑝−1‖𝑢_𝑚‖^𝑝_1,𝑝≤𝐶_𝜖‖𝑢_𝑚‖_1,𝑝+𝐶‖𝑢_𝑚‖^{𝛼−𝑝+3}_1,𝑝, (5.23)which give us0<𝑚₀^𝑝−1≤𝐶_𝜖‖𝑢_𝑚‖^𝑝−1_1,𝑝+𝐶‖𝑢_𝑚‖^{2𝑝−𝛼−3}_1,𝑝. (5.24)Since ‖𝑢_𝑚‖_1,𝑝→+∞ we arrive in0<𝑚₀^𝑝−1≤0, (5.25)which is impossible. Thus, (‖𝑢_𝑚‖_1,𝑝) is bounded. Consequently, up to a subsequence,‖𝑢_𝑚‖^𝑝_1,𝑝→𝑡₀,𝑢_𝑚⇀𝑢in𝑊₀^1,𝑝𝑢(Ω),_𝑚→𝑢in𝐿^𝑞(Ω),1≤𝑞<𝑝^∗,𝑢_𝑚(𝑥)→𝑢(𝑥)a.e.Ω. (5.26)And in view of continuity of 𝑀[𝑀⁺(‖𝑢_𝑚‖^𝑝_1,𝑝)]^𝑝−1→[𝑀⁺(𝑡₀)]^𝑝−1. (5.27)
We now fix 𝑙≤𝑚 and so 𝔹_𝑙⊂𝔹_𝑚. For 𝜑∈𝔹_𝑙 we have[𝑀⁺(‖𝑢_𝑚‖^𝑝_1,𝑝)]^𝑝−1_Ω|∇𝑢_𝑚|^𝑝−2∇𝑢_𝑚∇𝜑=_Ωℎ(𝑥)𝜑(𝜖+|𝑢_𝑚|)^{𝛾−𝑝+2}+_Ω|𝑢_𝑚|^{𝛼−𝑝+2}𝜑, (5.28)for all 𝜑∈𝔹_𝑙.
We now remark that|ℎ(𝑥)𝜑(𝜖+|𝑢_𝑚|)^{𝛾−𝑝+2}𝐶|≤𝜖^{𝛾−𝑝+2}|𝜑|∈𝐿¹(Ω),ℎ(𝑥)𝜑(𝜖+|𝑢_𝑚(𝑥)|)^{𝛾−𝑝+2}→ℎ(𝑥)𝜑(𝜖+|𝑢(𝑥)|)^{𝛾−𝑝+2}a.e.Ω. (5.29)By the Lebesgue dominated convergence theorem_Ωℎ(𝑥)𝜑(𝜖+|𝑢|)^{𝛾−𝑝+2}→_Ωℎ(𝑥)𝜑(𝜖+|𝑢|)^{𝛾−𝑝+2}. (5.30)Furthermore, because 𝑢_𝑚→𝑢∈𝐿^𝑝(Ω), we have|𝑢_𝑚|^{𝛼−𝑝+2}→|𝑢|^{𝛼−𝑝+2}in𝐿^{𝑝/(𝛼−𝑝+2)}(Ω), (5.31)and because Ω is a bounded domain and 𝑝<𝑝/(𝛼−𝑝+2) we have 𝐿^{𝑝/(𝛼−𝑝+2)}(Ω)⊂𝐿^𝑝(Ω) and so|𝑢_𝑚|^{𝛼−𝑝+2}→|𝑢|^{𝛼−𝑝+2}in𝐿^𝑝(Ω). (5.32)Consequently,|𝑢_𝑚(𝑥)|^{𝛼−𝑝+2}→|𝑢(𝑥)|^{𝛼−𝑝+2}a.e.inΩ, (5.33)and there is 𝑔∈𝐿^𝑝(Ω) such that|𝑢_𝑚(𝑥)|^{𝛼−𝑝+2}≤𝑔(𝑥)a.e.inΩ∀𝑚∈ℕ. (5.34)Thus,||𝑢_𝑚|^{𝛼−𝑝+2}𝜑|≤𝑔|𝜑|∈𝐿¹(Ω), (5.35)and using again the Lebesgue dominated convergence theorem_Ω|𝑢_𝑚|^{𝛼−𝑝+2}𝜑→_Ω|𝑢|^{𝛼−𝑝+2}𝜑∀𝜑∈𝔹_𝑙. (5.36)
We will now consider the following claim, whose proof will be postponed to Section 6.
Claim 1. The following convergence holds[𝑀⁺(‖𝑢_𝑚‖^𝑝_1,𝑝)]^𝑝−1_Ω|∇𝑢_𝑚|^𝑝−2∇𝑢_𝑚∇𝜑→[𝑀⁺(𝑡₀)]^𝑝−1_Ω|∇𝑢|^𝑝−2∇𝑢∇𝜑, (5.37)for all 𝜑∈𝑊₀^1,𝑝(Ω).
In the inequality (5.28) we make 𝑚→∞ and we use (5.27), (5.30), and (5.36) to obtain[𝑀⁺(𝑡₀)]^𝑝−1_Ω|∇𝑢|^𝑝−2∇𝑢∇𝜑=_Ωℎ(𝑥)𝜑(𝜖+|𝑢|)^{𝛾−𝑝+2}+_Ω|𝑢|^{𝛼−𝑝+2}, (5.38)for all 𝜑∈𝔹_𝑙.
Since 𝑙∈ℕ is arbitrary, the above inequality holds true for all 𝜑∈𝑊₀^1,𝑝(Ω). Because of this 𝑀⁺(𝑡₀)>0 and so 𝑀⁺(𝑡₀)=𝑀(𝑡₀), which implies[𝑀(𝑡₀)]^𝑝−1_Ω|∇𝑢|^𝑝−2∇𝑢∇𝜑=_Ωℎ(𝑥)𝜑(𝜖+|𝑢|)^{𝛾−𝑝+2}+_Ω|𝑢|^{𝛼−𝑝+2}, (5.39)for all 𝜑∈𝑊₀^1,𝑝(Ω).
Taking 𝑢 as a test function in the inequality (5.39), we get[𝑀(𝑡₀)]^𝑝−1‖𝑢‖^𝑝_1,𝑝=_Ωℎ(𝑥)𝑢(𝜖+|𝑢|)^{𝛾−𝑝+2}+_Ω|𝑢|^{𝛼−𝑝+2}𝑢. (5.40)
We now take 𝜑=𝑢_𝑚 in (5.28) to obtain[𝑀(‖𝑢_𝑚‖^𝑝_1,𝑝)]^𝑝−1‖𝑢_𝑚‖^𝑝_1,𝑝=_Ωℎ(𝑥)𝑢_𝑚(𝜖+|𝑢_𝑚|)^{𝛾−𝑝+2}+_Ω|𝑢_𝑚|^{𝛼−𝑝+2}𝑢_𝑚. (5.41)
Since|ℎ(𝑥)𝑢_𝑚(𝜖+|𝑢_𝑚|)^{𝛾−𝑝+2}𝐶|≤𝜖^{𝛾−𝑝+2}|𝑢_𝑚| (5.42)and because 𝑢_𝑚→𝑢∈𝐿^𝑞(Ω),1≤𝑞<𝑝^∗, it follows that |𝑢_𝑚(𝑥)|→|𝑢(𝑥)| a.e. in Ω and there is 𝑔∈𝐿¹(Ω) such that |𝑢_𝑚|≤𝑔 a.e. in Ω for all 𝑚∈ℕ. Consequently,|ℎ(𝑥)𝑢_𝑚(𝜖+|𝑢_𝑚|)^{𝛾−𝑝+2}𝐶|≤𝜖^{𝛾−𝑝+2}|𝑢_𝑚𝐶|≤𝜖^{𝛾−𝑝+2}𝑔∈𝐿¹(Ω). (5.43)
Reasoning as before, we obtain∫_Ωℎ(𝑥)𝑢_𝑚(𝜖+|𝑢_𝑚|)^{𝛾−𝑝+2}→∫_Ωℎ(𝑥)𝑢(𝜖+|𝑢|)^{𝛾−𝑝+2},∫_Ω|𝑢_𝑚|^{𝛼−𝑝+2}𝑢_𝑚→∫_Ω|𝑢|^{𝛼−𝑝+2}𝑢. (5.44)
Taking limits on both sides of the equality (5.41) and using ‖𝑢_𝑚‖^𝑝_1,𝑝→𝑡₀, (5.27), (5.44), we obtain[𝑀⁺(𝑡₀)]^𝑝−1𝑡₀=_Ωℎ(𝑥)𝑢(𝜖+|𝑢|)^{𝛾−𝑝+2}+_Ω|𝑢|^{𝛼−𝑝+2}𝑢. (5.45)
Comparing this last equality with the one in (5.40) we obtain 𝑡₀=‖𝑢‖^𝑝_1,𝑝 because 𝑀⁺(𝑡₀)>0. Then, from (5.39)[𝑀(‖𝑢‖^𝑝_1,𝑝)]^𝑝−1_Ω|∇𝑢|^𝑝−2∇𝑢∇𝜑=_Ωℎ(𝑥)𝜑(𝜖+|𝑢|)^{𝛾−𝑝+2}+_Ω|𝑢|^{𝛼−𝑝+2}𝜑 (5.46)for all 𝜑∈𝑊₀^1,𝑝(Ω).
This shows that 𝑢∈𝑊₀^1,𝑝(Ω) is a weak solution of the auxiliary problem (5.2) which is positive by virtue of the maximum principle. This proves the lemma.

In what follows, for each 𝑛∈ℕ, we set 𝜖=1/𝑛 and 𝑢_1/𝑛=𝑢_𝑛 where 𝑢_1/𝑛 is the solution obtained in the last lemma.

Lemma 5.4. There exists 𝛿>0 such that [𝑀(‖𝑢_𝑛‖^𝑝_1,𝑝)]^𝑝−1≥𝛿>0, for all 𝑛∈ℕ.

Proof. We will reason by contradiction. Suppose thatliminf[𝑀(‖𝑢_𝑛‖^𝑝_1,𝑝)]^𝑝−1=0. (5.47)If it is the case, the sequence (‖𝑢_𝑛‖^𝑝_1,𝑝) is bounded because, on the contrary, we would have ‖𝑢_𝑛‖^𝑝_1,𝑝>𝜃₁, perhaps for a subsequence, and so[𝑀(‖𝑢_𝑛‖^𝑝_1,𝑝)]^𝑝−1≥𝑚₀^𝑝−1>0, (5.48)and this would implyliminf[𝑀(‖𝑢_𝑛‖^𝑝_1,𝑝)]^𝑝−1≥𝑚₀^𝑝−1>0, (5.49)which is impossible in view of (5.47). Therefore, up to subsequences,‖𝑢_𝑛‖^𝑝_1,𝑝→𝜃₀,𝑢_𝑛⇀in𝑊₀^1,𝑝𝑢(Ω),_𝑛(𝑥)→𝑢(𝑥)a.e.inΩ. (5.50)Since 𝑀 is continuous, we have0=liminf[𝑀(‖𝑢_𝑛‖^𝑝_1,𝑝)]^𝑝−1=lim[𝑀(‖𝑢_𝑛‖^𝑝_1,𝑝)]^𝑝−1=[𝑀(𝜃₀)]^𝑝−1. (5.51)
We now note thatℎ(𝑥)(1/𝑛+𝑡)^{𝛾−𝑝+2}+𝑡^{𝛼−𝑝+2}≥ℎ(𝑥)(1+𝑡)^{𝛾−𝑝+2}+𝑡^{𝛼−𝑝+2}≥𝐶(1+𝑡)^{𝛾−𝑝+2}+𝑡^{𝛼−𝑝+2}≥𝐶^{0𝑥002𝑑𝑐}[1(1+𝑡)^{𝛾−𝑝+2}+𝑡^{𝛼−𝑝+2}]≥𝐶₀>0, (5.52)for all 𝑥∈Ω and 𝑡≥0, because the function 𝑡↦1/(1+𝑡)^{𝛾−𝑝+2}+𝑡^{𝛼−𝑝+2},𝑡≥0, attains a positive minimum.
Since−[𝑀(‖𝑢_𝑛‖^𝑝_1,𝑝)]^𝑝−1=ℎ(𝑥)(1/𝑛+𝑢_𝑛)^{𝛾−𝑝+2}+𝑢_𝑛^{𝛼−𝑝+2}≥𝐶₀>0,in𝑢Ω,_𝑛=0,on𝜕Ω. (5.53)Taking 𝜑∈𝐶¹₀(Ω),𝜑>0 in Ω, as a test function in the last equality, we obtain[𝑀(‖𝑢_𝑛‖^𝑝_1,𝑝)]^𝑝−1_Ω|∇𝑢_𝑛|^𝑝−2∇𝑢_𝑛∇𝜑≥𝐶_0Ω𝜑>0, (5.54)and so 𝑀(‖𝑢_𝑛‖^𝑝_1,𝑝)]^𝑝−1>0. Taking limit on the above expression we get0=[𝑀(𝜃₀)]^𝑝−1_Ω|∇𝑢|^𝑝−2∇𝑢∇𝜑≥𝐶_0Ω𝜑>0, (5.55)which is impossible. This completes the proof of the lemma.

Lemma 5.5. The sequence (‖𝑢_𝑛‖^𝑝_1,𝑝) is bounded, where 𝑢_𝑛 is as above.

Proof. First of all we note that[𝑀(‖𝑢_𝑛‖^𝑝_1,𝑝)]^𝑝−1‖𝑢_𝑛‖^𝑝_1,𝑝=_Ωℎ(𝑥)𝑢_𝑛(1/𝑛+𝑢_𝑛)^{𝛾−𝑝+2}+_Ω𝑢_𝑛^{𝛼−𝑝+3},_Ωℎ(𝑥)𝑢_𝑛(1/𝑛+𝑢_𝑛)^{𝛾−𝑝+2}≤𝐶₁‖𝑢_𝑛‖^{𝑝−𝛾−1}_1,𝑝,_Ω𝑢_𝑛^{𝛼−𝑝+3}≤𝐶₂‖𝑢_𝑛‖^{𝛼−𝑝+3}_1,𝑝, (5.56)where 𝐶₁ and 𝐶₂ are positive constants independent of 𝑛. Consequently,𝛿‖𝑢_𝑛‖^𝑝_1,𝑝≤[𝑀(‖𝑢_𝑛‖^𝑝_1,𝑝)]^𝑝−1‖𝑢_𝑛‖^𝑝_1,𝑝≤𝐶₁‖𝑢_𝑛‖^{𝑝−𝛾−1}_1,𝑝+𝐶₂‖𝑢_𝑛‖^{𝛼−𝑝+3}_1,𝑝. (5.57)Since 𝑝−𝛾−1<1,𝛼−𝑝+3<2≤𝑝, we conclude that (‖𝑢_𝑛‖^𝑝_1,𝑝) is bounded.

As a consequence of the preceding lemma we have0<𝛿≤[𝑀(‖𝑢_𝑛‖^𝑝_1,𝑝)]^𝑝−1≤𝑀_∞,∀𝑛∈ℕ. (5.58)

Lemma 5.6. The sequence (𝑢_𝑛), obtained in the last lemma, converges to a solution of problem (5.1).

Proof. As (𝑢_𝑛) is a bounded sequence in 𝑊₀^1,𝑝(Ω) we have, up to subsequence, that‖𝑢_𝑛‖^𝑝_1,𝑝→𝑡₀𝑢>0,_𝑛⇀𝑢in𝑊₀^1,𝑝𝑢(Ω),_𝑛→in𝐿^𝑞(Ω),1≤𝑞<𝑝^∗,𝑢_𝑛(𝑥)→𝑢(𝑥)a.e.Ω. (5.59)From the continuity of 𝑀,𝑀(‖𝑢_𝑛‖^𝑝_1,𝑝)→𝑀(𝑡₀). (5.60)
Let 𝜓₁>0 be an eigenfunction of (−Δ_𝑝,𝑊₀^1,𝑝(Ω)) associated to first eigenvalue 𝜆₁ and satisfying𝐶₀>𝜆₁𝑀_∞^𝑝−1𝜓₁^𝑝−1,∀𝑥∈Ω, (5.61)where 𝐶₀ is the constant obtained in (5.54). Consequently,−[𝑀(‖𝑢_𝑛‖^𝑝_1,𝑝)]^𝑝−1Δ_𝑝𝑢_𝑛=ℎ(𝑥)(1/𝑛+𝑢_𝑛)^{𝛾−𝑝+2}+𝑢_𝑛^{𝛼−𝑝+2}inΩ,≥𝐶₀>𝜆₁𝑀_∞^𝑝−1𝜓₁^𝑝−1in𝑢Ω,_𝑛=𝜓₁=0on𝜕Ω. (5.62)Since−[𝑀(‖𝑢_𝑛‖^𝑝_1,𝑝)]^𝑝−1Δ_𝑝𝑢_𝑛≥𝐶₀>𝜆₁𝑀_∞^𝑝−1𝜓₁^𝑝−1inΩ, (5.63)it follows that−Δ_𝑝([𝑀(‖𝑢_𝑛‖^𝑝_1,𝑝)]𝑢_𝑛)>−Δ_𝑝(𝑀_∞𝜓₁)inΩ, (5.64)and as [𝑀(‖𝑢_𝑛‖^𝑝_1,𝑝)]𝑢_𝑛=𝑀_∞𝜓₁=0 on 𝜕Ω. By the comparison principle,[𝑀(‖𝑢_𝑛‖^𝑝_1,𝑝)]𝑢_𝑛>𝑀_∞𝜓₁inΩ, (5.65)and from (5.58)𝑢_𝑛𝑀(𝑥)>^∞𝜓₁(𝑥)𝑀_∞^1/(𝑝−1)inΩ. (5.66)From which we conclude that 𝑢_𝑛(𝑥)↛0 for each 𝑥∈Ω.
Next, we will use the following.

Hardy-Sobolev Inequality. If 𝑢∈𝑊₀^1,𝑝(Ω) and 1<𝑝≤𝑁, then 𝑢/𝜓^𝑟₁∈𝐿^𝑟(Ω), where 1/𝑟=1/𝑝−(1−𝜏)/𝑁,0≤𝜏≤1, and‖𝑢𝜓^𝑟₁‖_𝐿𝑟≤‖∇𝑢‖_𝐿^𝑝, (5.67)where 𝐶>0 is a constant and is an eigenfunction of (−Δ_𝑝,𝑊₀^1,𝑝(Ω)) associated to the first eigenvalue 𝜆₁.

Whose proof may be found in [20].
As|ℎ(𝑥)𝜑(1/𝑛+𝑢_𝑛)^{𝛾−𝑝+2}|≤ℎ(𝑥)|𝜑||𝑢_𝑛|^{𝛾−𝑝+2}, (5.68)and using (5.66) it followsℎ(𝑥)|𝜑||𝑢_𝑛|^{𝛾−𝑝+2}<‖ℎ‖_∞𝐶|𝜑|𝜓₁^{𝛾−𝑝+2}. (5.69)From the Hardy-Sobolev inequality‖ℎ‖_∞𝐶|𝜑|𝜓₁^{𝛾−𝑝+2}∈𝐿¹(Ω). (5.70)Furthermore,ℎ(𝑥)𝜑(1/𝑛+𝑢_𝑛(𝑥))^{𝛾−𝑝+2}→ℎ(𝑥)𝜑𝑢(𝑥)^{𝛾−𝑝+2}inΩ, (5.71)and so, by the Lebesgue dominated convergence theorem_Ωℎ(𝑥)𝜑(1/𝑛+𝑢_𝑛(𝑥))^{𝛾−𝑝+2}→_Ωℎ(𝑥)𝜑𝑢(𝑥)^{𝛾−𝑝+2},∀𝜑∈𝑊₀^1,𝑝(Ω). (5.72)
Since Ω is bounded and 𝑝<𝑝/(𝛼−𝑝+2), we have 𝐿^{𝑝/(𝛼−𝑝+2)}↪𝐿^𝑝(Ω) and so𝑢_𝑛^{𝛼−𝑝+2}→𝑢^{𝛼−𝑝+2}in𝐿^𝑝(Ω). (5.73)Since Ω is bounded and 𝑝<𝑝/(𝛼−𝑝+2), we have 𝐿^{𝑝/(𝛼−𝑝+2)}↪𝐿^𝑝(Ω) and so𝑢_𝑛(𝑥)^{𝛼−𝑝+2}→𝑢(𝑥)^{𝛼−𝑝+2}a.e.inΩ, (5.74)and there exists 𝑔∈𝐿^𝑝(Ω) such that𝑢_𝑛^{𝛼−𝑝+2}≤𝑔a.e.inΩ,∀𝑛∈ℕ. (5.75)Hence,|𝑢_𝑛^{𝛼−𝑝+2}𝜑|≤𝑔|𝜑|∈𝐿¹(Ω), (5.76)and using again the Lebesgue dominated convergence theorem, we obtain_Ω𝑢_𝑛^{𝛼−𝑝+2}𝜑→_Ω𝑢^{𝛼−𝑝+2}𝜑, (5.77)for all 𝜑∈𝑊₀^1,𝑝(Ω).
As (𝑢_𝑛) is solution of the auxiliary problem[𝑀(‖𝑢_𝑛‖^𝑝_1,𝑝)]^𝑝−1_Ω|∇𝑢_𝑛|^𝑝−2∇𝑢_𝑛∇𝜑=_Ωℎ(𝑥)𝜑(1/𝑛+𝑢_𝑛)^{𝛾−𝑝+2}+_Ω𝑢_𝑛^{𝛼−𝑝+2}𝜑, (5.78)for all 𝜑∈𝑊₀^1,𝑝(Ω), taking limits on both sides of the above expression, using the same reasoning as in the proof of the claim and the convergences in (5.60), (5.72), and (5.77) we obtain[𝑀(𝑡₀)]^𝑝−1_Ω|∇𝑢|^𝑝−2∇𝑢∇𝜑=_Ωℎ(𝑥)𝜑𝑢^{𝛾−𝑝+2}+_Ω𝑢^{𝛼−𝑝+2}, (5.79)for all 𝜑∈𝑊₀^1,𝑝(Ω). Using 𝑢 as a test function in the last expression[𝑀(𝑡₀)]^𝑝−1‖𝑢‖^𝑝_1,𝑝=_Ωℎ(𝑥)𝑢^{𝑝−𝛾−1}+_Ω𝑢^{𝛼−𝑝+3}. (5.80)Taking 𝜑=𝑢_𝑛 in the expression (5.78), it follows that[𝑀(‖𝑢_𝑛‖^𝑝_1,𝑝)]^𝑝−1‖𝑢_𝑛‖^𝑝_1,𝑝=_Ωℎ(𝑥)𝑢_𝑛(1/𝑛+𝑢_𝑛)^{𝛾−𝑝+2}+_Ω𝑢_𝑛^{𝛼−𝑝+3}. (5.81)Noticing that 𝑢_𝑛→𝑢 in 𝐿^𝑞(Ω),1≤𝑞<𝑝^∗, and 1<1/(𝑝−𝛾−1) we have 𝐿^{1/(𝑝−𝛾−1)}(Ω)↪𝐿¹(Ω) and so𝑢_𝑛→𝑢in𝐿^{1/(𝑝−𝛾−1)}𝑢(Ω),_𝑛(𝑥)→𝑢(𝑥)a.e.inΩ, (5.82)and there exists 𝜔∈𝐿^{1/(𝑝−𝛾−1)}(Ω) such that0≤𝑢_𝑛(𝑥)≤𝜔(𝑥)a.e.inΩ,∀𝑛∈ℕ. (5.83)Thus,|ℎ(𝑥)𝑢_𝑛(1/𝑛+𝑢_𝑛)^{𝛾−𝑝+2}|≤‖ℎ‖_∞𝑢_𝑛^{𝑝−𝛾−1}∈𝐿¹(Ω),ℎ(𝑥)𝑢_𝑛(1/𝑛+𝑢_𝑛)^{𝛾−𝑝+2}→ℎ(𝑥)𝑢(𝑥)^{𝑝−𝛾−1}a.e.inΩ. (5.84)By the Lebesgue dominated convergence theorem_Ωℎ(𝑥)𝑢_𝑛(1/𝑛+𝑢_𝑛)^{𝛾−𝑝+2}→_Ωℎ(𝑥)𝑢(𝑥)^{𝑝−𝛾−1}, (5.85)and because 1<𝛼−𝑝+3<2, it follows that_Ω𝑢_𝑛^{𝛼−𝑝+3}→_Ω𝑢^{𝛼−𝑝+3}. (5.86) We now take limits on both sides of (5.81), by using (5.60), (5.85), and (5.86), to obtain[𝑀(𝑡₀)]^𝑝−1𝑡₀=_Ωℎ(𝑥)𝑢𝑢^{𝛾−𝑝+2}+_Ω𝑢^{𝛼−𝑝+3}, (5.87)that is,[𝑀(𝑡₀)]^𝑝−1𝑡₀=_Ωℎ(𝑥)𝑢^{𝑝−𝛾−1}+_Ω𝑢^{𝛼−𝑝+3}. (5.88)
Comparing (5.80), (5.88) and using [𝑀(𝑡₀)]^𝑝−1>0, we get ‖𝑢‖^𝑝_1,𝑝=𝑡₀. Therefore, from (5.79)[𝑀(‖𝑢‖^𝑝_1,𝑝)]^𝑝−1_Ω|∇𝑢|^𝑝−2∇𝑢∇𝜑=_Ωℎ(𝑥)𝜑𝑢^{𝛾−𝑝+2}+_Ω𝑢^{𝛼−𝑝+2}𝜑, (5.89)for all 𝜑∈𝑊₀^1,𝑝(Ω), which shows that 𝑢 is a weak solution of (5.1).

6. Proof of The Claim

We recall that we have to prove that[𝑀⁺(‖𝑢‖^𝑝_1,𝑝)]^𝑝−1_Ω|∇𝑢_𝑚|^𝑝−2∇𝑢_𝑚∇𝜑→[𝑀⁺(𝑡₀)]^𝑝−1_Ω|∇𝑢|^𝑝−2∇𝑢∇𝜑 (6.1)for all 𝜑∈𝑊₀^1,𝑝(Ω).

First of all, we claim that (−Δ_𝑝𝑢_𝑚)⊂(𝑊₀^1,𝑝(Ω))^′ is a bounded sequence. Indeed,|⟨−Δ_𝑝𝑢_𝑚,𝑣⟩|=|_Ω|∇𝑢_𝑚|^𝑝−2∇𝑢_𝑚∇𝑣|≤_Ω|∇𝑢_𝑚|^𝑝−1|∇𝑣|, (6.2)and because 𝑢_𝑚∈𝔹_𝑚⊂𝑊₀^1,𝑝(Ω), we have |∇𝑢_𝑚|∈𝐿^𝑝(Ω) and |∇𝑢_𝑚|^𝑝−1∈𝐿^{𝑝/(𝑝−1)}(Ω). Applying the Hölder inequality, by using the exponents 𝑝 and 𝑝/(𝑝−1), it follows that|⟨−Δ_𝑝𝑢_𝑚,𝑣⟩|≤(∫_Ω|∇𝑢_𝑚|^𝑝)^{(𝑝−1)/𝑝}(∫_Ω|∇𝑣|^𝑝)^1/𝑝=‖𝑢_𝑚‖^𝑝_1,𝑝‖𝑣‖_1,𝑝, (6.3)which implies‖−Δ_𝑝𝑢_𝑚‖_(𝑊0^1,𝑝_(Ω))′≤‖𝑢‖^𝑝_1,𝑝≤𝐶_𝜖, (6.4)where 𝐶_𝜖>0 is a constant, because the boundedness of (‖𝑢_𝑚‖_1,𝑝) was proved in Lemma 5.5.

Since 𝑊₀^1,𝑝(Ω) is a separable Banach space and (−Δ𝑢_𝑚) is a bounded sequence in (𝑊₀^1,𝑝(Ω))^′ then, up to a subsequence,−Δ_𝑝𝑢⇀_𝑚^∗𝜒in(𝑊₀^1,𝑝(Ω))^′, (6.5)that is,⟨−Δ_𝑝𝑢_𝑚,𝜓⟩→⟨𝜒,𝜓⟩∀𝜓∈𝑊₀^1,𝑝(Ω), (6.6)which is equivalent to_Ω|∇𝑢_𝑚|^𝑝−2∇𝑢_𝑚∇𝜓→⟨𝜒,𝜓⟩∀𝜓∈𝑊₀^1,𝑝(Ω). (6.7)

Since[𝑀⁺(‖𝑢_𝑚‖^𝑝_1,𝑝)]^𝑝−1→[𝑀⁺(𝑡₀)]^𝑝−1, (6.8)it follows from (6.7) that[𝑀⁺(‖𝑢_𝑚‖^𝑝_1,𝑝)]^𝑝−1_Ω|∇𝑢_𝑚|^𝑝−2∇𝑢_𝑚∇𝜓→[𝑀⁺(𝑡₀)]^𝑝−1⟨𝜒,𝜓⟩, (6.9)for all 𝜓∈𝑊₀^1,𝑝(Ω). Taking limits on both sides of (5.28) and using the convergences (5.30) and (5.36), we obtain[𝑀⁺(‖𝑢_𝑚‖^𝑝_1,𝑝)]^𝑝−1_Ω|∇𝑢_𝑚|^𝑝−2∇𝑢_𝑚∇𝜓→_Ωℎ(𝑥)𝜓(𝜖+|𝑢|)^{𝛾−𝑝+2}+_Ω|𝑢|^{𝛼−𝑝+2}𝜓, (6.10)for all 𝜓∈𝑊₀^1,𝑝(Ω). So, from (6.9), (6.10) and the uniqueness of the limit[𝑀⁺(𝑡₀)]^𝑝−1⟨𝜒,𝜓⟩=_Ωℎ(𝑥)𝜓(𝜖+|𝑢|)^{𝛾−𝑝+2}+_Ω|𝑢|^{𝛼−𝑝+2}𝜓, (6.11)for all 𝜓∈𝑊₀^1,𝑝(Ω).

Using the monotonicity of the operator (−Δ_𝑝), that is,⟨−Δ_𝑝𝜔−(−Δ_𝑝𝑣),𝜔−𝑣⟩≥0∀𝜔,𝑣∈𝑊₀^1,𝑝(Ω), (6.12)we have[𝑀⁺(‖𝜔‖^𝑝_1,𝑝)]^𝑝−1_Ω(|∇𝜔|^𝑝−2∇𝜔−|∇𝑣|^𝑝−2∇𝑣,∇𝜔−∇𝑣)𝑑𝑥≥0. (6.13)Taking 𝜔=𝑢_𝑚 and 𝑣=𝜓 in the last expression, we obtain[𝑀⁺(‖𝑢_𝑚‖^𝑝_1,𝑝)]^𝑝−1_Ω(|∇𝑢_𝑚|^𝑝−2∇𝑢_𝑚−|∇𝜓|^𝑝−2∇𝜓,∇𝑢_𝑚−∇𝜓)𝑑𝑥≥0, (6.14)and so[𝑀⁺(‖𝑢_𝑚‖^𝑝_1,𝑝)]^𝑝−1‖𝑢_𝑚‖^𝑝_1,𝑝−[𝑀⁺(‖𝑢_𝑚‖^𝑝_1,𝑝)]^𝑝−1∫_Ω|∇𝑢_𝑚|^𝑝−2∇𝑢_𝑚∇𝜓−[𝑀⁺(‖𝑢_𝑚‖^𝑝_1,𝑝)]^𝑝−1∫_Ω|∇𝜓|^𝑝−2∇𝜓∇𝑢_𝑚+[𝑀⁺(‖𝑢_𝑚‖^𝑝_1,𝑝)]^𝑝−1∫_Ω|∇𝜓|^𝑝≥0. (6.15)In (5.28) we take 𝜑=𝑢_𝑚 to obtain[𝑀⁺(‖𝑢_𝑚‖^𝑝_1,𝑝)]^𝑝−1‖𝑢_𝑚‖^𝑝_1,𝑝=_Ωℎ(𝑥)𝑢_𝑚(𝜖+|𝑢_𝑚|)^{𝛾−𝑝+2}+_Ω|𝑢_𝑚|^{𝛼−𝑝+2}𝑢_𝑚. (6.16)Consequently,∫_Ωℎ(𝑥)𝑢_𝑚(𝜖+|𝑢_𝑚|)^{𝛾−𝑝+2}+∫_Ω|𝑢_𝑚|^{𝛼−𝑝+2}𝑢_𝑚−[𝑀⁺(‖𝑢_𝑚‖^𝑝_1,𝑝)]^𝑝−1∫_Ω|∇𝑢_𝑚|^𝑝−2∇𝑢_𝑚∇𝜓−[𝑀⁺(‖𝑢_𝑚‖^𝑝_1,𝑝)]^𝑝−1∫_Ω|∇𝜓|^𝑝−2∇𝜓∇𝑢_𝑚+[𝑀⁺(‖𝑢_𝑚‖^𝑝_1,𝑝)]^𝑝−1∫_Ω|∇𝜓|^𝑝≥0, (6.17)using the definition of −Δ_𝑝∫_Ωℎ(𝑥)𝑢_𝑚(𝜖+|𝑢_𝑚|)^{𝛾−𝑝+2}+∫_Ω|𝑢_𝑚|^{𝛼−𝑝+2}𝑢_𝑚−[𝑀⁺(‖𝑢_𝑚‖^𝑝_1,𝑝)]^𝑝−1⟨−Δ_𝑝𝑢_𝑚,𝜓⟩−[𝑀⁺(‖𝑢_𝑚‖^𝑝_1,𝑝)]^𝑝−1⟨−Δ_𝑝𝜓,𝑢_𝑚⟩+[𝑀⁺(‖𝑢_𝑚‖^𝑝_1,𝑝)]^𝑝−1⟨−Δ_𝑝𝜓,𝜓⟩≥0, (6.18)taking limits as 𝑚→∞ and using (5.44) we have∫_Ωℎ(𝑥)𝑢(𝜖+|𝑢|)^{𝛾−𝑝+2}+∫_Ω|𝑢|^{𝛼−𝑝+2}𝑢−[𝑀⁺(‖𝑢‖^𝑝_1,𝑝)]^𝑝−1⟨𝜒,𝜓⟩−[𝑀⁺(𝑡₀)]^𝑝−1⟨−Δ_𝑝𝜓,𝑢⟩+[𝑀⁺(𝑡₀)]^𝑝−1⟨−Δ_𝑝𝜓,𝜓⟩≥0. (6.19)We note that[𝑀⁺(𝑡₀)]^𝑝−1⟨𝜒,𝜓⟩=_Ωℎ(𝑥)𝜓(𝜖+|𝑢|)^{𝛾−𝑝+2}+_Ω|𝑢|^{𝛼−𝑝+2}, (6.20)for all 𝜓∈𝔹_𝑙⊂𝑊₀^1,𝑝(Ω), and so[𝑀⁺(𝑡₀)]^𝑝−1⟨𝜒,𝑢⟩=_Ωℎ(𝑥)𝑢(𝜖+|𝑢|)^{𝛾−𝑝+2}+_Ω|𝑢|^{𝛼−𝑝+2}𝑢. (6.21)Therefore,[𝑀⁺(𝑡₀)]^𝑝−1⟨𝜒,𝑢⟩−[𝑀⁺(𝑡₀)]^𝑝−1⟨𝜒,𝜓⟩−[𝑀⁺(𝑡₀)]^𝑝−1⟨−Δ_𝑝𝜓,𝑢⟩+[𝑀⁺(𝑡₀)]^𝑝−1⟨−Δ_𝑝𝜓,𝜓⟩≥0, (6.22)that is,⟨𝜒−(−Δ_𝑝𝜓),𝑢−𝜓⟩≥0. (6.23)Setting𝜓=𝑢−𝜆𝜑,𝜆>0,𝜑∈𝔹_𝑙, (6.24)we obtain⟨𝜒,−[−Δ_𝑝(𝑢−𝜆𝜑)],𝜑⟩≥0,∀𝜆>0,𝜑∈𝔹_𝑙, (6.25)taking limits on both sides of the above expression as 𝜆→0 and using the continuity of the operator −Δ_𝑝⟨𝜒,−(−Δ_𝑝𝑢),𝜑⟩≥0,∀𝜑∈𝔹_𝑙. (6.26)So,⟨𝜒−(−Δ_𝑝𝑢),−𝜑⟩≥0, (6.27)and we obtain⟨𝜒−(−Δ_𝑝𝑢),𝜑⟩≤0,∀𝜑∈𝔹_𝑙. (6.28)Hence,⟨𝜒−(−Δ_𝑝𝑢),𝜑⟩=0∀𝜑∈𝔹_𝑙, (6.29)which implies 𝜒=−Δ_𝑝𝑢, because 𝑙 is arbitrary. This means that the above equality is valid for all 𝜑∈𝑊₀^1,𝑝(Ω). From (6.10) and (6.11) we conclude that[𝑀⁺(‖𝑢_𝑚‖^𝑝_1,𝑝)]^𝑝−1_Ω|∇𝑢_𝑚|^𝑝−2∇𝑢_𝑚∇𝜓→[𝑀⁺(𝑡₀)]^𝑝−1⟨−Δ𝑢,𝜓⟩, (6.30)for all 𝜓∈𝑊₀^1,𝑝(Ω). Hence,[𝑀⁺(‖𝑢_𝑚‖^𝑝_1,𝑝)]^𝑝−1_Ω|∇𝑢_𝑚|^𝑝−2∇𝑢_𝑚∇𝜓→[𝑀⁺(𝑡₀)]^𝑝−1_Ω|∇𝑢|^𝑝−2∇𝑢∇𝜓 (6.31)for all 𝜓∈𝑊₀^1,𝑝(Ω). This concludes the proof of the claim.

Acknowledgment

The author is partially supported by CNP-q-Brazil-301603/2007-3 and FADESP.