On the Weak Solution of a Semilinear Boundary Value Problem without the Landesman-Lazer Condition

Abstract

We prove the existence of weak solution to a semilinear boundary value problem without the Landesman-Lazer condition.

1. Introduction

We consider the nonlinear boundary value problem Δ𝑢+𝜆_𝑘𝑢𝑢+=𝑔0(𝑢)o=nℎ𝜕Ω(𝑥,)inΩ,((11..12)) where Ω⊂ℝ^𝑛 is open and bounded, ℎ∈𝐿²(Ω), 𝜆_𝑘 is a simple eigenvalue of −Δ corresponding to the eigenvector 𝜙_𝑘, and the nonlinearity 𝑔∶ℝ→ℝ satisfies the following conditions: ||𝑔(𝑢)−𝑔(𝑣)||≤𝐿|𝑢−𝑣|(Lipschitzcontinuity)forsomeconstant𝐿>0.(H)

Landesman and Lazer [1] considered the problem (1.1)-(1.2) with continuous function 𝑔 satisfying 𝑔(−∞)<𝑔(𝜉)<𝑔(∞), where 𝑔(±∞)=lim_𝑠→±∞𝑔(𝑠) exist and are finite. The authors showed that if 𝜙_𝑘 is an eigenfunction corresponding to 𝜆_𝑘, Ω⁺={𝑥∈Ω∶𝜙_𝑘>0} and Ω⁻={𝑥∈Ω∶𝜙_𝑘<0}, then the necessary and sufficient condition for the existence of weak solution of (1.1)-(1.2) is that𝑔(−∞)_Ω+𝜙_𝑘𝑑𝑥+𝑔(∞)_Ω−𝜙_𝑘𝑑𝑥<_Ωℎ𝜙_𝑘𝑑𝑥<𝑔(∞)_Ω+𝜙_𝑘𝑑𝑥+𝑔(−∞)_Ω−𝜙_𝑘𝑑𝑥.(1.3) The condition (1.3) is the well-known Landesman-Lazer condition, named after the authors. The result of the paper [1] has since been generalized by a number of authors which include [2–9], to mention a few.

We mention, briefly, few works without the assumption of the Landesman-Lazer condition. The perturbation of a second order linear elliptic problems by nonlinearity without Landesman-Lazer condition was investigated in [10]. The function 𝑔(𝑢) was assumed to be a bounded continuous function satisfying 𝑔(𝑡)𝑡≤0,𝑡∈ℝ.(1.4) The nonhomogeneous term ℎ was assumed to be an 𝐿^∞-function orthogonal to an eigenfunction 𝜙 in 𝐿², which corresponds to a simple eigenvalue 𝜆₁. Ha [11] considered the solvability of an operator equation without the Landesman-Lazer condition. The author used a nonlinear Carathéodory function 𝑔(𝑥,𝑢) which satisfies the conditions ||𝑔𝑢(𝑔𝑥,(𝑢𝑥),||𝑢≤)≥𝑏(0𝑥,),(1.5) for almost all 𝑥∈Ω and all 𝑢∈ℝ, where 𝑏∈𝐿²(Ω). The solvability of the operator equation is proved under some hypotheses on 𝑔(𝑥,𝑢). The nonhomogeneous term ℎ was assumed to be an 𝐿²-function. Iannacci and Nkashama proved existence of solutions to a class of semilinear two-point eigenvalue boundary value problems at resonance without the Landesman-Lazer condition, by imposing the same conditions as in [11] in conjunction with some other hypotheses on 𝑔 and ℎ. Furthermore, the existence of solution was proved only for the eigenvalue 𝜆=1. Assuming a Carathéodory function 𝑓(𝑥,𝑢) with some growth restriction and assuming an 𝐿²-function ℎ, Santanilla [12] proved existence of solution to a nonlinear eigenvalue boundary value problem (for eigenvalue 𝜆=1) without Landesman-Lazer condition. Du [13] proved the existence of solution for nonlinear second-order two-point boundary value problems, by allowing the eigenvalue 𝜆 of the problem to change near the eigenvalues of 𝑚²𝜋² of the problem 𝑦^+𝑚²𝜋²𝑦=0,𝑦(0)=𝑦(1)=0. The author did not use the Landesman-Lazer condition and imposed weaker conditions on 𝑔(𝑢) than in [12]. Recently, Sanni [14] proved the existence of solution to the same problem considered by Du [13] with 𝜆=𝑚²𝜋² exactly, without assuming the Landesman-Lazer condition. The author assumed that |𝑔^(𝑢)|≤𝐶=constant and ℎ∈𝐿²(0,1). Other works without the assumption of Landesman-Lazer condition include [15–21]. We mention that most of the papers on this topic use the methods in [22] and [12]. The method of upper and lower solutions is used in [14]. For several other related resonance problems, we refer the reader to the book of Rădulescu [23].

The current work constitutes further deductions on the problem considered by Landesman and Lazer [1] and is motivated by previous works and by asking if it is possible to obtain a weak solution of (1.1)-(1.2) by setting 𝑢∶=𝜙_𝑘𝑣(𝑥). The answer is in the affirmative. The substitution gives rise to a degenerate semilinear elliptic equation. Consequently, we prove the existence of weak solution to the degenerate semilinear elliptic equation in a 𝜙^2𝑘-weight Sobolev’s space, by using the Schaefer’s fixed point theorem. For information on weighted Sobolev’s spaces, the reader is referred to [24, 25]. The current work is significant in that the condition H enables a relaxation of the Landesman-Lazer condition (1.3), and the solution 𝑢 to (1.1)-(1.2) is constructed using the eigenfunctions 𝜙_𝑘. Furthermore, the current analysis takes care of the situation where 𝑔(∞)=𝑔(−∞)=0.

The remaining part of this paper is organized as follows: the weighted Sobolev’s spaces used are defined in Section 2. In addition, we use the substitution 𝑢=𝜙_𝑘𝑣 to get the degenerate semilinear elliptic equation in 𝑣, from which we give a definition of a weak solution. Furthermore, we state two theorems used in the proof of the existence result. In Section 3, we prove the existence and uniqueness of solution to an auxiliary linear problem. In Section 4, we prove a necessary condition for the existence of solution to (1.1)-(1.2) before proving the existence of solution to (1.1)-(1.2). At the end of Section 4, we prove that 𝑢∶=𝜙_𝑘𝑣 is in 𝐻¹⁰(Ω), provided that 𝑣∈𝑋. Finally, we give an illustrative example in Section 5 for which our result applies.

2. Preliminaries

We define the following weighted Sobolev’s spaces used in this paper:𝐿²Ω,𝜙^2𝑘∶=𝑤∶Ω⟶ℝsuchthat‖𝑤‖_𝐿²_(Ω,𝜙^2𝑘₎<∞,(2.1) where ‖𝑤‖_𝐿²_(Ω,𝜙^2𝑘₎=∫_Ω𝜙^2𝑘𝑤²𝑑𝑥.𝐻¹Ω,𝜙^2𝑘∶=𝑤∶Ω⟶ℝsuchthat‖𝑤‖_𝐻¹_(Ω,𝜙^2𝑘₎<∞,(2.2) where ‖𝑤‖_𝐻¹_(Ω,𝜙^2𝑘₎=∫_Ω𝜙^2𝑘𝑤²𝑑𝑥+∫_Ω𝜙^2𝑘|∇𝑤|²𝑑𝑥.

For brevity, we set 𝑋=𝐻¹(Ω,𝜙^2𝑘).

Set 𝑢∶=𝜙_𝑘𝑣(𝑥) in (1.1) to deduce −Δ𝜙_𝑘+𝜆_𝑘𝜙_𝑘𝑣−𝜙_𝑘Δ𝑣−2∇𝜙_𝑘⋅∇𝑣=𝑔𝜙_𝑘𝑣−ℎ(𝑥)inΩ.(2.3) Note that the first term on the left of (2.3) vanishes, multiply (2.3) by 𝜙_𝑘 and use (1.2) to deduce −∇⋅𝜙^2𝑘∇𝑣𝜙=𝜙_𝑘𝑔𝜙_𝑘𝑣−𝜙_𝑘ℎ(𝑥)inΩ,_𝑘𝑣=0on𝜕Ω.(2.4) Thus, if we can prove the existence of solution to (2.4), then 𝑢∶=𝜙_𝑘𝑣 solves (1.1)-(1.2). Indeed, we will prove that the solution 𝑢 belongs to the Sobolev space 𝐻¹⁰(Ω).

Definition 2.1. We say that 𝑣∈𝑋 is a weak solution of the problem (2.4) provided _Ω𝜙^2𝑘∇𝑣⋅∇𝜁𝑑𝑥=_Ω𝜙_𝑘𝜁𝑔𝜙_𝑘𝑣𝑑𝑥−_Ω𝜙_𝑘𝜁ℎ𝑑𝑥,(2.5) for each 𝜁∈𝑋.

Definition 2.2. Let 𝑋 be a Banach space and 𝐴∶𝑋→𝑋 a nonlinear mapping. 𝐴 is called compact provided for each bounded sequence {𝑢_𝑘}^∞𝑘=1 the sequence {𝐴[𝑢_𝑘]}^∞𝑘=1 is precompact; that is, there exists a subsequence {𝑢_𝑘𝑗}^∞𝑗=1 such that {𝐴[𝑢_𝑘𝑗]}^∞𝑗=1 converges in 𝑋 (see [26]).
The following theorems are applied in this paper.

Theorem 2.3 (Bolzano-Weierstrass). Every bounded sequence of real numbers has a convergent subsequence (see [27]).

Theorem 2.4 (Schaefer’s Fixed Point Theorem). Let 𝑋 be a Banach space and 𝐴∶𝑋⟶𝑋(2.6) a continuous and compact mapping. Suppose further that the set {𝑢∈𝑋∣𝑢=𝜏𝐴[𝑢]forsome0≤𝜏≤1}(2.7) is bounded. Then 𝐴 has a fixed point (see [26]).

3. Auxiliary Linear Problem

Consider the linear boundary value problem: 𝐿𝑣∶=−∇⋅𝜙^2𝑘∇𝑣+𝜙𝜇𝜙^2𝑘𝑣=𝜇𝜙^2𝑘𝑠+𝜙_𝑘𝑔𝜙_𝑘𝑠−𝜙_𝑘ℎinΩ,(3.1)_𝑘𝑣=0on𝜕Ω,(3.2) where 𝜇 is a strictly positive constant; 𝑠∈𝐿²(Ω,𝜙^2𝑘), 𝑔(𝜙_𝑘𝑠), and ℎ are functions of 𝑥 only.

Theorem 3.1 (a priori estimates). Let 𝑣 be a solution of (3.1)-(3.2). Then 𝑣∈𝑋 and we have the estimate ‖𝑣‖^2𝑋≤𝐶‖𝑠‖^2𝐿_{2(Ω,𝜙2𝑘)}+‖ℎ‖^2𝐿_2(Ω)+1<∞,(3.3) for some appropriate constant 𝐶>0.

Proof. Multiply (3.1) by 𝑣, integrate by parts and apply (3.2) to get _Ω𝜙=^2𝑘||∇𝜇𝑣||²_Ω𝑑𝑥+𝜇_Ω𝜙^2𝑘𝑣²𝑑𝑥≤𝜇_Ω𝜙^2𝑘𝑣𝑠𝑑𝑥+_Ω𝑣𝜙_𝑘𝑔𝜙_𝑘𝑠𝑑𝑥−_Ω𝑣𝜙_𝑘ℎ𝑑𝑥𝜙^2𝑘𝑣²𝑑𝑥^1/2_Ω𝜙^2𝑘𝑠²𝑑𝑥^1/2+_Ω𝜙^2𝑘𝑣²𝑑𝑥^1/2_Ω||𝑔𝜙_𝑘𝑠||²𝑑𝑥^1/2+_Ω𝜙^2𝑘𝑣²𝑑𝑥^1/2_Ω≤3𝜖_Ωℎ²𝑑𝑥^1/2(byHölder'sinequality)𝜙^2𝑘𝑣²𝑑𝑥+14𝜖𝜇²_Ω(b𝜙y^2𝑘𝑠C²𝑑au𝑥c+hy'_Ωs||i𝑔ne𝜙qu_𝑘a𝑠li||t²y𝑑𝑥wi+th_Ω𝜖)ℎ.²𝑑𝑥(3.4) Using 𝐻, the second term in the bracket on the right side of (3.4) may be estimated as ||𝑔𝜙_𝑘𝑠||𝑔−𝜙𝑔(0)||²≤𝐿²||𝜙_𝑘𝑠||²or_𝑘𝑠||²≤−||𝑔(0)||²+2||𝑔𝜙_𝑘𝑠||||𝑔(0)||+𝐿²||𝜙_𝑘𝑠||²≤−||𝑔(0)||²+12||𝑔𝜙^𝑘𝑠||²+2||𝑔(0)||²+𝐿²||𝜙_𝑘𝑠||²(byYoung'sinequality).(3.5) Simplifying (3.5), we deduce ||𝑔𝜙_𝑘𝑠||≤𝐶1+||𝜙_𝑘𝑠||,(3.6) (see [26]) for some constant 𝐶=𝐶(𝐿,|𝑔(0)|). Notice that (3.6) implies that _Ω||𝑔𝜙_𝑘𝑠||²𝑑𝑥≤𝐶1+‖𝑠‖_𝐿²_(Ω,𝜙^2𝑘₎²<∞,(3.7) so that 𝑔(𝜙_𝑘𝑠)∈𝐿²(Ω).
Using (3.7) and choosing 𝜖>0 sufficiently small in (3.4) and simplifying, we deduce (3.3).

Definition 3.2. (i) The bilinear form 𝐵[⋅,⋅] associated with the elliptic operator 𝐿 defined by (3.1) is 𝐵[𝑣,𝜁]∶=_Ω𝜙^2𝑘∇𝑣⋅∇𝜁𝑑𝑥+𝜇_Ω𝜙^2𝑘𝑣𝜁𝑑𝑥,(3.8) for 𝑣,𝜁∈𝑋,
(ii) 𝑣∈𝑋 is called a weak solution of the boundary value problem (3.1)-(3.2) provided 𝐵[𝑢,𝜁]=𝜇𝜙^2𝑘𝑠+𝜙_𝑘𝑔𝜙_𝑘𝑠−𝜙_𝑘ℎ,𝜁,(3.9) for all 𝜁∈𝑋, where (⋅,⋅) denotes the inner product in 𝐿²(Ω).

Theorem 3.3. 𝐵[𝑢,𝑣] satisfies the hypotheses of the Lax-Milgram theorem precisely. In other words, there exists constants 𝛼,𝛽 such that(i)|𝐵[𝑣,𝜁]|≤𝛼‖𝑣‖_𝑋‖𝜁‖_𝑋, (ii)𝛽‖𝑣‖^2𝑋≤𝐵[𝑣,𝑣], for all 𝑣,𝜁∈𝑋.

Proof. We have ||𝐵[𝑣,𝜁]||=||||_Ω𝜙^2𝑘∇𝑣⋅∇𝜁𝑑𝑥+𝜇_Ω𝜙^2𝑘𝑣𝜁𝑑𝑥||||≤𝜇_Ω𝜙^2𝑘𝑣²𝑑𝑥^1/2_Ω𝜙^2𝑘𝜁²𝑑𝑥^1/2+_Ω𝜙^2𝑘||∇𝑣||²𝑑𝑥^1/2_Ω≤𝛼‖𝑣‖_𝑋𝜙^2𝑘||∇𝜁||²𝑑𝑥^1/2byHölder^sinequality‖𝜁‖_𝑋,(3.10) for appropriate constant 𝛼>0. This proves (i).
We now proof (ii). We readily check that 𝛽‖𝑣‖^2𝑋≤_Ω𝜙^2𝑘||∇𝑣||²𝑑𝑥+𝜇_Ω𝜙^2𝑘𝑣²𝑑𝑥=𝐵[𝑣,𝑣],(3.11) for some constant 𝛽>0. We can for example take 𝛽=min{1,𝜇}.

Theorem 3.4. There exists unique weak solution to the degenerate linear boundary value problem (3.1)-(3.2).

Proof. The hypothesis on ℎ and (3.7) imply that 𝑔(𝜙_𝑘𝑠)−ℎ∈𝐿²(Ω). For fixed 𝑔(𝜙_𝑘𝑠)−ℎ, set ⟨𝜇𝜙^2𝑘𝑠+𝜙_𝑘𝑔(𝜙_𝑘𝑠)−𝜙_𝑘ℎ,𝜁⟩∶=(𝜇𝜙^2𝑘𝑠+𝜙_𝑘𝑔(𝜙_𝑘𝑠)−𝜙_𝑘ℎ,𝜁)_𝐿²_(Ω) for all 𝜁∈𝑋 (where ⟨,⋅,⟩ denotes the pairing of 𝑋 with its dual). This is a bounded linear functional on 𝐿²(Ω) and thus on 𝑋. Lax-Milgram theorem (see, e.g., [26]) can be applied to find a unique function 𝑣∈𝑋 satisfying 𝐵[𝑣,𝜁]=𝜇𝜙^2𝑘𝑠+𝜙_𝑘𝑔𝜙_𝑘𝑠−𝜙_𝑘ℎ,𝜁,(3.12) for all 𝜁∈𝑋. Consequently, 𝑣 is the unique weak solution of the problem (3.1)-(3.2).

4. Main Results

Theorem 4.1. The necessary condition that 𝑢∈𝐻¹⁰(Ω) be a weak solution to (1.1)-(1.2) is that _Ω𝑔(𝑢)𝜙_𝑘𝑑𝑥=_Ωℎ𝜙_𝑘𝑑𝑥.(4.1)

Proof. Suppose 𝑢∈𝐻¹⁰(Ω) is a weak solution of (1.1)-(1.2). For a test function 𝜙_𝑘, using integration by parts, we have: _ΩΔ𝑢𝜙_𝑘𝑑𝑥+𝜆_𝑘_Ω=−_Ω𝑢𝜙_𝑘𝑑𝑥+_Ω𝑔(𝑢)𝜙_𝑘𝑑𝑥∇𝑢⋅∇𝜙_𝑘+𝜆_𝑘_Ω=𝑢𝜙_𝑘𝑑𝑥+_Ω𝑔(𝑢)𝜙_𝑘𝑑𝑥_Ω𝑢Δ𝜙_𝑘+𝜆_𝑘𝜙_𝑘𝑑𝑥+_Ω𝑔(𝑢)𝜙_𝑘𝑑𝑥=_Ωℎ𝜙_𝑘𝑑𝑥,(4.2) from which (4.1) follows, since Δ𝜙_𝑘+𝜆_𝑘𝜙_𝑘=0.

Theorem 4.2. Let the condition (4.1) of Theorem 4.1 holds. Then there exists a weak solution to the problem (2.4).

Proof. The proof is split in seven steps.
Step 1. A fixed point argument to (2.4) is −∇⋅𝜙^2𝑘∇𝑤+𝜇𝜙^2𝑘𝑤𝜙=𝜇𝜙^2𝑘𝑣+𝜙_𝑘𝑔𝜙_𝑘𝑣−𝜙_𝑘ℎ(𝑥)inΩ,_𝑘𝑤=0on𝜕Ω.(4.3) Define a mapping 𝐴∶𝑋⟶𝑋(4.4) by setting 𝐴[𝑣]=𝑤 whenever 𝑤 is derived from 𝑣 via (4.3). We claim that 𝐴 is a continuous and compact mapping. Our claim is proved in the next two steps.Step 2. Choose 𝑣,𝑣̃∈𝑋, and define 𝐴[𝑣]=𝑤,𝐴[𝑣]̃=𝑤. For two solutions 𝑤,𝑤∈𝑋 of (4.3), we have −∇⋅𝜙^2𝑘∇𝑤−𝑤+𝜇𝜙^2𝑘𝑤𝜙−𝑤=𝜇𝜙^2𝑘(𝑣−𝑣)̃+𝜙_𝑘𝑔𝜙_𝑘𝑣−𝜙_𝑘𝑔𝜙_𝑘𝑣̃inΩ,_𝑘𝑤−𝑤=0on𝜕Ω.(4.5) Using (4.5), we obtain an analogous estimate to (3.4), namely: _Ω𝜙_𝑘||∇𝑤−∇𝑤||²𝑑𝑥+𝜇_Ω𝜙^2𝑘||𝑤−𝑤||²≤+3𝜖1_Ω𝜙^2𝑘||𝑤−𝑤||²𝑑𝑥4𝜖𝜇²_Ω𝜙_𝑘||𝑣−𝑣||̃^2𝐿_{2(Ω,𝜙2𝑘)}+_Ω||𝑔𝜙_𝑘𝑣−𝑔𝜙_𝑘𝑣̃||²𝑑𝑥.(4.6) Now _Ω||𝑔𝜙_𝑘𝑣−𝑔𝜙_𝑘𝑣̃||²𝑑𝑥≤_Ω𝜙^2𝑘𝐿²||𝑣−𝑣||̃²𝑑𝑥,(4.7) using the condition (H). We may now use (4.7) in (4.6) and simplify to deduce ‖𝐴[𝑣]−𝐴[𝑣]̃‖_𝑋=‖‖𝑤−𝑤‖‖_𝑋≤𝐶‖𝑣−𝑣‖̃_𝐿²_(Ω,𝜙^2𝑘₎≤𝐶‖𝑣−𝑣‖̃_𝑋,(4.8) for some constant 𝐶>0. Thus, the mapping 𝐴 is Lipschitz continuous, and hence continuous.Step 3. Let {𝑣_𝑘}^∞𝑘=1 be a bounded sequence in 𝑋. By Bolzano-Weierstrass theorem, it has a convergent subsequence, say {𝑣_𝑘𝑗}^∞𝑗=1. Define 𝑣∶=lim_{𝑘𝑗→∞}𝑣_𝑘𝑗.(4.9) Using (4.8)-(4.9), we deduce lim_𝑘𝑗→‖‖𝐴𝑣_𝑘𝑗−𝐴[𝑣]‖‖_𝑋≤lim_{𝑘𝑗→∞}𝐶‖‖𝑣_𝑘𝑗−𝑣‖‖_𝑋=0.(4.10) Thus, 𝐴[𝑣_𝑘𝑗]→𝐴[𝑣] in 𝑋. Therefore, 𝐴 is compact.Step 4. Define a set 𝐾∶={𝑝∈𝑋∶𝑝=𝜏𝐴[𝑝]forsome0≤𝜏≤1}. We will show that 𝐾 is a bounded set. Let 𝑣∈𝐾. Then 𝑣=𝜏𝐴[𝑣] for some 𝜏∈[0,1]. Thus, we have 𝑣/𝜏=𝐴[𝑣]. By the definition of the mapping 𝐴, 𝑤=𝑣/𝜏 is the solution of the problem −∇⋅𝜙^2𝑘∇𝑣𝜏+𝜇𝜙^2𝑘𝑣𝜏=𝜇𝜙^2𝑘𝜙𝑣+𝜙_𝑘𝑔𝜙_𝑘𝑣−𝜙_𝑘ℎ(𝑥)inΩ,_𝑘𝑣𝜏=0on𝜕Ω.(4.11) Now, (4.11) are equivalent to −∇⋅𝜙^2𝑘∇𝑣+𝜇𝜙^2𝑘𝑣=𝜙𝜇𝜏𝜙^2𝑘𝑣+𝜏𝜙_𝑘𝑔𝜙_𝑘𝑣−𝜏𝜙_𝑘ℎ(𝑥)inΩ,_𝑘𝑣=0on𝜕Ω.(4.12) Using (4.12) we have an analogous estimate to (3.3) of Theorem 3.1, namely: ‖𝑣‖^2𝑋≤𝜏𝐶‖𝑣‖^2𝐿_{2(Ω,𝜙2𝑘)}+‖ℎ‖^2𝐿_2(Ω)+1.(4.13) Choosing 𝜏∈[0,1] sufficiently small in (4.13) and simplifying, we conclude that ‖𝑣‖_𝑋≤𝐶‖ℎ‖^2𝐿_2(Ω)+1<∞(4.14) for some constant 𝐶>0. Equation (4.14) implies that the set 𝐾 is bounded, since 𝑣 was arbitrarily chosen.
Since the mapping 𝐴 is continuous and compact and the set 𝐾 is bounded, by Schaefer’s fixed point theorem (see, e.g., [26]), the mapping 𝐴 has a fixed point in 𝑋.Step 5. Write 𝜙_𝑘𝑣₀=𝜙_𝑘𝑣|_𝜕Ω=0. For 𝑚=0,1,2,…, inductively define 𝑣_𝑚+1∈𝑋 to be the unique weak solution of the linear boundary value problem −∇⋅𝜙^2𝑘∇𝑣_𝑚+1+𝜇𝜙^2𝑘𝑣𝜙_𝑚+1=𝜇𝜙^2𝑘𝑣_𝑚+𝜙_𝑘𝑔𝜙_𝑘𝑣_𝑚−𝜙_𝑘ℎ(𝑥)inΩ,(4.15)_𝑘𝑣_𝑚+1=0on𝜕Ω.(4.16) Clearly, our definition of 𝑣_𝑚+1∈𝑋 as the unique weak solution of (4.15)-(4.16) is justified by Theorem 3.4. Hence, by the definition of the mapping 𝐴, we have for 𝑚=0,1,2,…∶𝑣_𝑚+1=𝐴𝑣_𝑚.(4.17) Since 𝐴 has a fixed point in 𝑋, there exists 𝑣∈𝑋 such that lim_𝑚→∞𝑣_𝑚+1=lim_𝑚→∞𝐴𝑣_𝑚=𝐴[𝑣]=𝑣.(4.18)Step 6. Using (4.15)-(4.16), we obtain an analogous estimate to (3.3), namely: ‖‖𝑣_𝑚+1‖‖^2𝑋≤≤𝐶𝐶‖‖‖‖𝑣𝑣_𝑚𝑚‖‖^2𝐿_{2(Ω,𝜙2𝑘)}+‖ℎ‖^2𝐿_2(Ω)+1‖‖^2𝑋+‖ℎ‖^2𝐿_2(Ω)+1(4.19) for some appropriate constant 𝐶>0. Using (4.18), we take the limit on the right side of (4.19) to deduce that sup_𝑚‖‖𝑣_𝑚‖‖_𝑋<∞.(4.20) Equation (4.20) implies the existence of a subsequence {𝑣_𝑚𝑗}^∞𝑗=1 converging weakly in 𝑋 to 𝑣∈𝑋.
Furthermore, using (3.7), we deduce _Ω||𝑔𝜙_𝑘𝑣_𝑚||²𝑑𝑥≤𝐶1+‖‖𝑣_𝑚‖‖^2𝐿_{2(Ω,𝜙2𝑘)}².(4.21) Again, we use (4.18) to obtain the limit on the right side of (4.21) to deduce that sup_𝑚‖‖𝑔𝜙_𝑘𝑣_𝑚‖‖_𝐿²_(Ω)<∞.(4.22) Equation (4.22) implies the existence of a subsequence {𝑔(𝜙_𝑘𝑣_𝑚𝑗)}^∞𝑗=1 converging weakly in 𝐿²(Ω) to 𝑔(𝜙_𝑘𝑣) in 𝐿²(Ω).Step 7. Finally, we verify that 𝑣 is a weak solution of (2.4). For brevity, we take the subsequences of the last step as {𝑣_𝑚}^∞𝑚=1 and {𝑔(𝜙_𝑘𝑣_𝑚)}^∞𝑚=1. Fix 𝜁∈𝑋. Multiply (4.15) by 𝜁, integrate by parts and apply (4.16) to get _Ω𝜙^2𝑘∇𝑣_𝑚+1⋅∇𝜁𝑑𝑥+𝜇_Ω𝜙^2𝑘𝑣_𝑚+1𝜁𝑑𝑥=𝜇_Ω𝜙^2𝑘𝑣_𝑚𝜁𝑑𝑥+_Ω𝜁𝜙_𝑘𝑔𝜙_𝑘𝑣_𝑚𝑑𝑥−_Ω𝜁𝜙_𝑘ℎ𝑑𝑥.(4.23) Using the deductions of the last step, we let 𝑚→∞ in (4.23) to obtain _Ω𝜙_𝑘∇𝑣⋅∇𝜁𝑑𝑥+𝜇_Ω𝜙^2𝑘𝑣𝜁𝑑𝑥=𝜇_Ω𝜙^2𝑘𝑣𝜁𝑑𝑥+_Ω𝜁𝜙_𝑘𝑔𝜙_𝑘𝑣𝑑𝑥−_Ω𝜁𝜙_𝑘ℎ𝑑𝑥,(4.24) from which canceling the terms in 𝜇, we obtain (2.5) as desired.

Theorem 4.3. Let 𝑣∈𝑋 be the solution of (3.1)-(3.2). Then, the solution 𝑢∶=𝜙_𝑘𝑣 of (1.1)-(1.2) belongs to 𝐻¹⁰(Ω), and we have the estimate ‖𝑢‖_𝐻¹⁰_(Ω)≤𝐶‖𝑣‖_𝑋,(4.25) for some constant 𝐶>0.

Proof. We split the proof in two steps.
Step 1. Recall that 𝜙_𝑘 satisfies the equations: Δ𝜙_𝑘+𝜙𝜆_𝑘𝜙_𝑘=0inΩ∈ℝ^𝑛,(4.26)_𝑘=0on𝜕Ω.(4.27) Multiplying (4.26) by 𝑣²𝜙_𝑘, integrating by parts and applying (4.27) we compute _Ω𝑣²𝜙_𝑘Δ𝜙_𝑘𝑑𝑥+𝜆_𝑘_Ω𝑣²𝜙^2𝑘𝑑𝑥=0or_Ω∇𝑣²𝜙_𝑘⋅∇𝜙_𝑘𝑑𝑥=𝜆_𝑘_Ω𝑣²𝜙^2𝑘𝑑𝑥or_Ω||∇𝜙_𝑘||²𝑣²𝑑𝑥=𝜆_𝑘_Ω𝑣²𝜙^2𝑘𝑑𝑥−2_Ω𝜙_𝑘𝑣∇𝑣⋅∇𝜙_𝑘𝑑𝑥(4.28)≤𝜆_𝑘_Ω𝑣²𝜙^2𝑘𝑑𝑥+𝜖_Ω||∇𝜙_𝑘||²𝑣²𝑑𝑥+1𝜖_Ω𝜙^2𝑘||∇𝑣||²𝑑𝑥,(4.29) by Cauchy’s inequality with 𝜖. Choosing 𝜖>0 sufficiently small in (4.29) and simplifying, we deduce _Ω||∇𝜙_𝑘||²𝑣²𝑑𝑥≤𝐶‖𝑣‖^2𝑋,(4.30) for some constant 𝐶>0.Step 2. We have _Ω𝑢²𝑑𝑥=_Ω𝜙^2𝑘𝑣²𝑑𝑥,_Ω||∇𝑢||²𝑑𝑥=≤2_Ω_Ω||∇𝜙_𝑘𝑣||²𝑑𝑥=_Ω||∇𝜙_𝑘𝑣+𝜙_𝑘∇𝑣||²𝑑𝑥||∇𝜙_𝑘||²𝑣²𝑑𝑥+2_Ω𝜙^2𝑘||∇𝑣||²𝑑𝑥≤𝐶‖𝑣‖^2𝑋,(using(4.30))(4.31) for some constant 𝐶>0. Thus, 𝑢∈𝐻¹(Ω). Hence, by a Sobolev’s embedding theorem (see [26, page 269]), we have that 𝑢∈𝐻¹⁰(Ω), since 𝑢|_𝜕Ω=0.

5. Illustrative Example

Consider the following special case for 𝑛=1: 𝑢^+𝑢𝑢(−0)2=𝑢=𝑢(1𝜋)i=n0(.0,𝜋),(5.1) In this case, the eigenfunction 𝜙_𝑘=sin𝑥, 𝑔(𝑢)=−2𝑢, and ℎ=1. Clearly 𝑔(𝑢) is Lipschitz continuous and ℎ∈𝐿²(Ω). Provided the necessary condition−2^𝜋0𝑢sin𝑥𝑑𝑥=^𝜋0sin𝑥𝑑𝑥(5.2) is satisfied; Theorems 4.2 and 4.3 ensure the existence of a solution 𝑢∶=𝜙_𝑘𝑣(𝑥)∈𝐻¹⁰(Ω) of the problem (5.1). Now, the problem (5.1) admits the solution𝑢=sinh(𝜋−𝑥)+sinh𝑥sinh𝜋−1.(5.3) Using (5.3) in (5.2), it is not difficult to verify that the necessary condition −2^𝜋0sinh(𝜋s−in𝑥h)𝜋+sinh𝑥−1sin𝑥𝑑𝑥=^𝜋0sin𝑥𝑑𝑥=2(5.4) is satisfied.