# Multiplicity Results for thepx-Laplacian Equation with Singular Nonlinearities and Nonlinear Neumann Boundary Condition

Authors: K. Saoudi; M. Kratou; S. Alsadhan
Journal: International Journal of Differential Equations
 (2016)

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## Abstract

We investigate the singular Neumann problem involving thep(x)-Laplace operator: Pλ{-Δpxu+|u|px-2u  =1/uδx+fx,u, in  Ω;u>0,inΩ;∇upx-2∂u/∂ν=λuqx,on∂Ω}, where Ω⊂RNN≥2 is a bounded domain with C2 boundary, λ is a positive parameter, and px,qx,δx, and fx,u are assumed to satisfy assumptions(H0)–(H5) in the Introduction. Using some variational techniques, we show the existence of a number Λ∈0,∞ such that problem Pλ has two solutions for λ∈0,Λ, one solution for λ=Λ, and no solutions for λ>Λ.


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## Body

## 1. Introduction

The purpose of this paper is to study the existence of multiple solutions for the following inhomogeneous singular Neumann problem involving thepx-Laplace operator:(1)Pλ-Δpxu+upx-2u=1uδx+fx,u,inΩ,u>0,inΩ,∇upx-2∂u∂ν=λuqx,on⁡∂Ω.HereΩ⊂RNN≥2 is a bounded domain with C2 boundary; λ is a positive parameter. For any continuous and bounded function a we define a+≔sup⁡ax and a-≔inf⁡a(x). Associated with problem Pλ we have the singular functional Eλ:W1,pxΩ→R given by(2)Eλu≝∫Ω∇upxpxdx+∫Ωupxpxdx-∫Ωu1-δx1-δxdx-∫ΩFx,udx-λ∫∂Ωuqx+1qx+1dx, where Fx,t=∫0tf(x,s)ds.Definition 1.
u ∈ W 1 , p x Ω is called a generalized solution of the equation(3)-Δpxu+upx-2u=1uδx+fx,uif for all φ∈C0∞(Ω)(4)∫Ω∇upx-2∇u∇φdx+∫Ωupx-2uφdx=∫Ωu-δxφdx+∫Ωfx,uφdx+λ∫∂Ωuqxφdx.Obviously, every weak solution of problem(Pλ) is also a generalized solution of (3).The operatorΔpxu≔div⁡∇upx-2∇u is called px-Laplace where p is a continuous nonconstant function. This differential operator is a natural generalization of the p-Laplace operator Δpu≔div⁡∇up-2∇u, where p>1 is a real constant. However, the px-Laplace operator possesses more complicated nonlinearity than p-Laplace operator, due to the fact that Δpx is not homogeneous. This fact implies some difficulties; for example, we cannot use the Lagrange Multiplier Theorem in many problems involving this operator.The study of differential and partial differential equations involving variable exponent is a new and an interesting topic. The interest in studying such problems was stimulated by their applications in elastic mechanics, fluid dynamics, electrorheological fluids, image processing, flow in porous media, calculus of variations, nonlinear elasticity theory, heterogeneous porous media models (see Acerbi and Mingione [1], Diening [2]), and so forth. These physical problems were facilitated by the development of Lebesgue and Sobolev spaces with variable exponent.At this point, we briefly recall related existence and multiplicity results for elliptic equations with Neumann boundary conditions. Neumann type problems are studied in [3–6] and references therein. The multiplicity result for Neumann problem with Sobolev critical nonlinearity has been studied in [5] where authors considered the problem(5)-Δu+u=up,u>0inΩ,∂u∂ν=λψuqon∂Ω.Here Ω⊂RN, N≥3, and 0<q<1<p≤2N/N-2. They proved the existence of Λ~ such that problem (5) admits at least two solutions for all λ∈0,Λ~, one solution when λ=Λ~, and no solutions for λ>Λ~. The problem in two dimensions has been considered in [6] where the authors extended the results obtained by [5].Results forpx-Laplacian problems Neumann boundary conditions are rare (see [7, 8]). In [7], Fan and Deng studied the Neumann problems with px-Laplace operator and the nonlinear potential fx,u under appropriate assumptions. By using the subsupersolution method and variation method, the authors get the multiplicity of positive solutions. In [8], Sreenadh and Tiwari extend previous works on nonlinear parametric problems with the px-Laplace operator to the case where the Neumann boundary condition is nonlinear. Precisely, under supplementary hypotheses on p,q, the authors show that there exists a finite number Λ such that the posed problem has two solutions for λ∈(0,Λ), one solution for λ=Λ, and no solutions for λ>Λ.Before stating our main results, we make the following assumptions throughout this paper:(H0)
p·∈CΩ¯, 1<p-≤p+<N, and p+≤p∗-. As usual, p∗x=Npx/N-px.(H1)
0<qx∈Cβ∂Ω for some β∈0,1 satisfying 0≤q++1<p-p-/p+.(H2)
0<δx∈CΩ¯ satisfying 0<δ-≤δ+<1.Let f:Ω×R→R be a nondecreasing Carathéodory function satisfying the following: (H3)
fx,0=0 and fx,tt≥0 for all x,t∈Ω×R.(H4)
There existC1,C2>0 such that, for γ>0, ax∈CΩ¯ with p+-1<γ and p+<a-. (6)C1uγ≤fx,t≤C2tax-1,∀x∈Ω,t>0.(H5)
There exist a constantM>0 and θ>p+ such that(7)0<θFx,t≤fx,tt,∀x∈Ω,t>M.Next we describe in a more precise way our main results.Theorem 2.
Assume that(H0)–(H5) hold and ax≤p∗x=Np(x)/N-px in (6). Then there exists Λ>0 with the following properties:(1)
ProblemPλ has a solution uλ for every λ∈0,Λ.(2)
ProblemPλ has a solution if λ=Λ.(3)
ProblemPλ does not have any solution if λ>Λ.Theorem 3.
Assume that(H0)–(H5) hold and ax<p∗x in (6). Then, problem Pλ has at least two distinct solutions uλx and vλx for every λ∈0,Λ.This paper is organized as follows. In Section2, we will recall some basic facts about the variable exponent Lebesgue and Sobolev spaces which we will use later. Proofs of our results will be presented in Sections 3 and 5.


## 2. Generalized Lebesgue-Sobolev Spaces Setting

To deal with thepx-Laplacian problem, we need to introduce some functional spaces Lp·Ω, W1,p·Ω, W01,p·Ω and properties of the px-Laplacian which we will use later. Denote by SΩ the set of all measurable real-valued functions defined in Ω. Note that two measurable functions are considered as the same element of SΩ when they are equal almost everywhere. Let (8)Lp·Ω=u∈SΩ:∫Ωuxpxdx<∞,with the norm(9)up·=uLp·Ω=inf⁡λ>0:∫Ωuxλpxdx≤1. The space Lp·Ω,·p· becomes a Banach space. We call it variable exponent Lebesgue space. Moreover, this space is a separable, reflexive, and uniform convex Banach space; see [9, Theorems 1.6, 1.10, and 1.14].The variable exponent Sobolev space(10)W1,p·Ω=u∈Lp·Ω:∇u∈Lp·Ω can be equipped with the norm(11)u=up·+∇up·,∀u∈W1,p·Ω. Note that W01,p·Ω is the closure of C0∞Ω in W1,p·Ω under the norm u=∇up·. The spaces W1,p·Ω and W01,p·Ω are separable, reflexive, and uniform convex Banach spaces (see [9, Theorem 2.1]). The inclusion between Lebesgue spaces also generalizes naturally: if 0<Ω<∞ and p1, p2 are variable exponents so that p1x≤p2x almost everywhere in Ω then there exists the continuous embedding Lp2xΩ↪Lp1xΩ.We denote byLqxΩ the conjugate space of LpxΩ, where 1/qx+1/px=1. For u∈LpxΩ and v∈LqxΩ, the Hölder type inequality(12)∫Ωuxvxdx≤1p-+1q-upxvqxholds true.An important role in manipulating the generalized Lebesgue spaces is played by the modular of theLpxΩ space, which is the mapping ρpx:LpxΩ→R defined by (13)ρpxu=∫Ωupxdx. If un,u∈LpxΩ and p+<∞, then the following relations hold true.Lemma 4.
Consider the following:(14)uLpx>1⟹uLpxp-≤ρpxu≤uLpxp+,uLpx<1⟹uLpxp+≤ρpxu≤uLpxp-,un-uLpx⟶0iffρpxun-u⟶0.The following result generalizes the well-known Sobolev embedding theorem.Theorem 5 (see [10, 11]).
LetΩ⊂RN be an open bounded domain with Lipschitz boundary and assume that p∈CΩ¯ with px>1 for each x∈Ω¯. If r∈CΩ¯ and px≤rx≤p∗x for all x∈Ω¯, then there exists a continuous embedding W1,pxΩ↪LrxΩ. Also, the embedding is compact rx<p∗x almost everywhere in Ω¯, where(15)p∗x=NpxN-px,ifpx<N,+∞,ifpx≥N.Now, we recall the following boundary trace embedding theorem from [12].Theorem 6.
LetΩ⊂RN be an open bounded domain with Lipschitz boundary. If q∈C∂Ω such that (16)1≤qx≤N-1pxN-px,∀x∈∂Ω, then W1,px↪Lqx∂Ω.Next we give a comparison principle as follows.Lemma 7 (see [8, Lemma 3.2]).
Letu,v∈W1,p(·)(Ω) be nonnegative functions satisfying(17)-Δpxu+upx-1≥-Δpxv+vpx-1,x∈Ω,∇upx-2∂u∂ν≥∇vpx-2∂v∂ν. Then u≥v almost everywhere in Ω.We recall the following strong maximum principle from [13].Theorem 8.
Letu,v∈C1,βΩ¯, for some 0<β<1, satisfy 0≨u, 0≨v and(18)-Δpxu-λuδx=gx≥hx=-Δpxv-λvδx,with u=v=0 on ∂Ω, where g,h∈L∞Ω are such that 0≤g<h pointwise everywhere in Ω. If(19)∂u∂n>0,∂v∂n>0,on∂Ω, where n is the inward unit normal on ∂Ω, then, the following strong comparison principle holds: (20)u>v,inΩ;thereisapositiveϵsuchthat∂u-v∂n≥ϵ,on∂Ω.


## 3. Existence of a Solution

In this section, we show the existence of a local minimum forEλ in a small neighborhood of the origin in W1,pxΩ. Firstly, let us define(21)Λ≝sup⁡λ>0:Pλhas  a  weak  solution.Lemma 9.
There existsλ0>0 such that Pλ admits a solution for λ<λ0.Proof.
Using (14) and the embeddings in Theorem 5, we estimate Eλu for uW1,pxΩ=r as follows:(22)Eλu=∫Ω∇upxpxdx+∫Ωupxpxdx-∫Ωu1-δx1-δxdx-∫ΩFx,udx-λ∫∂Ωuqx+1qx+1dx≥∇uLpxΩp++uLpxΩp+-uLpxΩ1-δ+-uLpxΩa--λuLpx∂Ωq-+1≥12p+-1∇uLpxΩ+uLpxΩp+-uW1,pxΩ1-δ+-uW1,pxΩa--λuW1,pxΩq-+1=12p+-1uW1,pxΩp+-uW1,pxΩ1-δ+-uLpxΩa--λuW1,pxΩq-+1.
Hence, noting that0<δ+<1 and 1-δ+<p+<a-, we can choose r0>0 small enough, and there exists δ0>0 such that(23)∫Ω∇upxpxdx+∫Ωupxpxdx-∫ΩFx,udx-λ∫∂Ωuqx+1qx+1dx≥0,∀u∈Br0,∫Ω∇upxpxdx+∫upxpxdx-∫ΩFx,udx-λ∫∂Ωuqx+1qx+1dx≥δ0,∀u∈∂Br0.Moreover, since W1,p(x)(Ω)↪L1-δ(x)(Ω), we have, for λ>0, small enough, (24)Eλ∖∂Br0≥δ02>0. Set (25)c0=infu∈Br0⁡Eλu.Now, note that, for t→0+ and u∈W1,pxΩ, we have(26)Eλtu=∫Ωtpx∇upxpxdx+∫Ωtupxpxdx-∫Ωtu1-δx1-δxdx-∫ΩFx,tudx-λ∫∂Ωtuqx+1qx+1dx≤tp-p+∫Ω∇upxdx+tp-p+∫Ωupxdx-t1-δ+1-δ+∫Ωu1-δxdx-∫tu≥t∗Fx,tudx-λtq-+1q-+1∫∂Ωuqx+1dx≤tp-p+∫Ω∇upxdx+tp-p+∫Ωupxdx-t1-δ+1-δ+∫Ωu1-δxdx-tθt∗θ∫tu≥t∗Fx,tudx-λtq-+1q-+1∫∂Ωuqx+1dx≤tp-p+∫Ω∇upxdx+tp-p+∫Ωupxdx-t1-δ+1-δ+∫Ωu1-δxdx-tθ∫u≥t∗Fx,t∗dx-λtq-+1q-+1∫∂Ωuqx+1dx.
Since1-δ+<p+<θ, this implies that Eλtu→-∞ as t→∞. Thus, inf∥u∥W1,pxΩ≤r⁡Eλu=c0<0. Now, let uj⊂Br0 be a minimizing sequence for c0. Then ujW1,pxΩ<r0 for r0∈0,r. Now by the Ekeland variational principle, there exists a sequence vj such that vjW1,p(x)(Ω)<r and(27)uj-vjW1,pxΩ≤1n,Eλvj≤Eλuj,Eλ′vj⟶0inW1,pxΩ∗,vj⟶uλweakly  inW1,pxΩ.Moreover, using the Brézis-Lieb lemma in [14] combined with the p(x) version of Theorem 2.1 in Boccardo and Murat [15], it follows that ∇vj(x)→∇uλ(x) for almost every x∈Ω,(28)∫Ω∇vjpxpxdx=∫Ω∇vj-uλpxpx+∫Ω∇uλpxpx+oj1,∫Ωfx,vjdx=∫Ωfx,uλdx+∫Ωfx,vj-uλdx+oj1,∫Ωvj1-δx1-δxdx=∫Ωuλ1-δx1-δxdx+∫Ωvj-uλ1-δx1-δxdx+oj1.Hence, from (28) and the compact boundary trace embedding in Theorem 5, we get Eλvj≥Eλuλ+oj1. Thus, it follows that Eλuλ=c0<0. Hence uλ≢0 and it is a local minimizer of Eλ in W1,pxΩ.We prove now the existence of positive solution to(Pλ) for 0<λ<Λ. Precisely, we have the following result.Lemma 10.
Problem(Pλ) possesses a solution uλ for 0<λ<Λ.Proof.
Fix0<λ<λ2<Λ. λ2 such that there exist solutions to (Pλ) for λ=λ2, say uλ2. Note that uλ2 is a supersolution for (Pλ). It is clear that 0 is not a local minimizer of Eλ on 0,uλ2∩W1,p(x)(Ω) since Eλ(0)=0 and Eλ(tv)<0 for t→0+. Now, we show the existence of a local minimizer of the functional energy. For this, we use the cut-off argument. Define (29)f~x,s=fx,0,ifs<0,fx,s,if0≤s≤uλ2,fx,uλ2,ifs>uλ2x,g~x,s=0,ifs<0,s-δx,if0≤s≤uλ2,uλ2-δx,ifs>uλ2x,h~x,s=0,ifs<0,sqx,if0≤s≤uλ2,uλ2qx,ifs>uλ2x. Also define the functional E~λ:W1,pxΩ→R by(30)E~λu=∫Ω∇upxpxdx+∫Ωupxpxdx-∫ΩG~x,udx-∫ΩF~x,udx-λ∫∂ΩH~x,udx,where F~(x,t)=∫0tf~x,sds, G~(x,t)=∫0tg~x,sds, and H~x,t=∫0th~x,sds. From the dominated convergence and the compact boundary trace embedding in Theorem 5, it is easy to see that E~λ is bounded below and is weakly lower semicontinuous in W1,p(x)(Ω). Then, there exists u~λ∈W1,p(x)(Ω) such that E~λ achieves its global minimum in W1,p(x)(Ω). Moreover, since E~λ is C1(W1,p(x)(Ω),R) from Lemma A.4, u~λ solves the equation(31)-Δpxu~λ+u~λpx-2u~λ=g~x,u~λ+f~x,u~λ,inΩ,∇u~λpx-2∂u~λ∂ν=λh~x,u~λ,on∂Ω.
Now, using the strong maximum principle (see Theorem8) u~λ≢0 and u~λ>0 since 0 is not a local minimizer of Eλ on 0,uλ2∩W1,pxΩ. By the definition of f~,g~, and h~ we have(32)g~x,0≤g~x,u~λ≤g~x,uλ2,∀x∈Ω,f~x,0≤f~x,u~λ≤f~x,uλ2,∀x∈Ω,h~x,0≤h~x,u~λ≤h~x,uλ2,∀x∈Ω.Again, by the strong comparison principle (see Lemma 7), we conclude that u~λ≤uλ2 in Ω and ∂/∂νuλ2-u~λ≤0 on ∂Ω. Hence u~λ is a solution to (Pλ). This completes the proof of Lemma 10.Now, we show the following result.Lemma 11.
There exists at least one positive weak solution forλ=Λ to problem (Pλ).Proof.
Letλkk∈N∈0,Λ, λk↑Λ as k→∞, and uλk be a solution of Pλk such that uλk-1≤uλk for all x∈Ω. Now, taking uλk as test function in Pλk, we get(33)∫Ω∇uλkpxdx+∫Ωuλkpxdx=∫Ωuλk1-δxdx+∫Ωfx,uλkuλkdx+λk∫∂Ωuλkqx+1dx.Moreover, as Eλkuλk≤Eλk0=0, we have(34)∫Ω∇uλkpxpxdx+∫Ωuλkpxpxdx-∫Ωuλk1-δx1-δxdx-∫ΩFx,uλkdx-λk∫∂Ωuλkqx+1qx+1dx≤0.It follows that(35)1p+∫Ω∇uλkpxdx+∫Ωuλkpxdx≤∫Ωuλk1-δxdx+∫ΩFx,uλkdx+λk∫∂Ωuλkqx+1dx.Now, from (7) there exists C>0 such that, for θ>p+ and for all t>0, (36)θFx,t≤tfx,t+C.Moreover, using Theorem 4.2 in [16], we get the existence of the constants C1>0, C2>0 such that(37)C1dxθ1≤u_λx≤C2dxθ2,where θ1=maxd(x)≤σ⁡px/px-1+δx,θ2=mind(x)≤σ⁡px/px-1+δx, and dx=dist⁡x,∂Ω. On the other hand, we recall the following inequality due to Lieberman [17]. There exists a constant K(Ω)>0 such that(38)∫∂Ωu≤∫Ω∇u+KΩ∫Ωu,∀u∈W1,1Ω.
Inserting (36), (37), and (38) in (33), we get supk∈N⁡uλkW1,pxΩ<∞. It follows that uλk is bounded in W1,p(x)(Ω) since uλk(x)≥uλ1(x) for all x∈Ω. Without loss of generality, uλk⇀uΛ in W1,p(x)(Ω) and then by the Sobolev imbedding uλk⇀uΛ in Lq(x)(Ω) and uλk(x)→uΛ(x) for a.e. x∈Ω. By the L∞(Ω)-regularity results of [13], the boundedness of uλk implies the boundedness of uλkL∞Ω. By the C1,αΩ-regularity Theorem 16, the boundedness of uλkL∞Ω implies the boundedness of uλkC1,αΩ¯, where α∈0,1 is a constant. Thus, we have uλk→uΛ in C1Ω¯. For every v∈W1,pxΩ, since uλk is a solution of problem Pλk, we have that, for each k,(39)∫Ω∇uλkpx-2∇uλk∇vdx+∫Ωuλkpx-2uλkvdx=∫Ωuλk-δxvdx+∫Ωfx,uλkvdx+λk∫∂Ωuλkqxvdx.Passing to the limit in (39) as k→∞ yields (40)∫Ω∇uΛpx-2∇uΛ∇vdx+∫ΩuΛpx-2uΛvdx=∫ΩuΛ-δxvdx+∫Ωfx,uΛvdx+Λ∫∂ΩuΛqxvdx, which shows that uΛ is a solution of PΛ. Obviously uΛ≥0 and uΛ≢0. Hence uΛ is a positive solution of PΛ in W1,pxΩ. This completes the proof of Lemma 11.Then we prove the following nonexistence result.Lemma 12.
Consider the following:
Λ < ∞.Proof.
Letuλ be a solution of (Pλ). Taking φ≡1 as a test function in the weak formulation of (Pλ), we get(41)∫Ωuλpx-1dx=∫Ωuλ-δxdx+∫Ωfx,uλdx+λ∫∂Ωuλqxdx.On the other hand, we have(42)∫Ωuλpx-1≤∫Ωuλp+-1+C1.Using assumption(H2) we have C2∫Ωuλ-δ+≤∫Ωu-δ(x), and using (6) we have C3∫Ωuλγ≤∫Ωfx,udx. Therefore, from (41) and (42) we get(43)∫Ωuλγ≤C3∫Ωuλp+-C4∫Ωuλ-δ++C5.Now, since p+-1<γ and -δ+<γ, by the embedding of Lγ(Ω) into Lp+-1(Ω) and by the embedding of Lγ(Ω) into L-δ+(Ω) we obtain(44)∫Ωuλp+-1≤C6∫Ωuλγp+-1/γ,∫Ωuλ-δ+≤C7∫Ωuλγ-δ+/γ.Substituting (44) in (43) we get (45)uλLγΩγ≤C8+C9uλLγΩp+-1+C10uλLγΩ-δ+. Hence, uλLγΩ is bounded by a constant independent of λ. Now, taking φ=uλ-q- as a test function in the weak formulation of (Pλ) and using(H2) we get(46)∫Ωuλpx-1-q-dx≥∫Ωuλ-δx-q-dx+∫Ωfx,uλuλ-q-dx+λ∫∂Ωuλqx-q-dx≥λ∫∂Ωuλqx-q-dx≥λ∫Γuλq--q-dx=λΓN-1.Now since(47)∫Ωuλpx-1-q-dx≤C+uλLγΩγit follows that Λ is finite. The proof of Lemma 12 is now completed.Proof of Theorem2.
Theorem2 follows from Lemmas 10, 11, and 12.Now, we prove that the solution of problem(Pλ) obtained in Lemma 10 is a local minimum for the functional energy associated to problem (Pλ). Precisely, we have the following result.Lemma 13.
Letuλ be the weak solution of problem Pλ obtained in Lemma 10. Then, uλ is a local minimum for Eλ∖C1Ω¯.Proof.
Fix0<λ1<λ<λ2<Λ and let uλ1,uλ2 be solutions to Pλ for λ=λ1 and λ=λ2, respectively, such that uλ1≤uλ≤uλ2. By Lemma 7,  uλ1<uλ<uλ2 on Ω¯. Define the following cut-off functions: (48)f^x,t=fx,uλ1x,t<uλ1x,fx,t,uλ1x≤t≤uλ2x,fx,uλ2x,t>uλ2x,x∈Ω,t∈R,g^x,t=uλ1x-δx,t<uλ1x,t-δx,uλ1x≤t≤uλ2x,uλ2-δx,t>uλ2x,x∈Ω,t∈R,h^x,t=ψxuλ1qx,t<uλ1x,ψxtqx,uλ1x≤t≤uλ2x,ψxuλ2qx,t>uλ2x,x∈∂Ω,t∈R. Then the corresponding functional E^λ:W1,pxΩ→R given by(49)E^λu=∫Ω∇upxpx+∫Ωupxpx-∫ΩG^x,u-∫ΩF^x,u-λ∫∂ΩH^x,u, where F^x,u=∫0uf^(x,t)dtx∈Ω, G^(x,u)=∫0ug^x,tdt(x∈Ω), and H^(x,u)=∫0uh^x,tdtx∈∂Ω. Firstly, from Lemma A.4  E^λ is C1W1,pxΩ,R. Then, it is simple to see that E^λ is coercive and bounded below. Let uλ denote the global minimum of E^λ on W1,pxΩ which satisfies the equation(50)-div⁡∇upx-2∇u+upx-2u=g^x,u+f^x,u,u>0inΩ,∇upx-2∂u∂ν=λh^x,u,on⁡∂Ω.Therefore, from the regularity results Theorem 16, we conclude that uλ is in C1,α(Ω¯) for some 0<α<1. Now, using(H2) and since f(x,t) is nondecreasing, by the definition of f^, g^, and h^ we have (51)g^x,uλ1≤g^x,uλ≤g^x,uλ2,∀x∈Ω,f^x,uλ1≤f^x,uλ≤f^x,uλ2,∀x∈Ω,h^x,uλ1≤h^x,uλ≤h^x,uλ2,∀x∈∂Ω. Again, by the strong comparison principle (see Lemma 7), we conclude that uλ1≤uλ≤uλ2 in Ω. Let β=infx∈Ω¯⁡uλ1-uλ2+∇uλ1-∇uλ2. We claim that β>0. If not, then there exists x∈∂Ω such that uλ1=uλ2 and ∇uλ1=∇uλ2. But this contradicts the boundary data as λ1≠λ2. Therefore, if v∈C=u∈C1Ω¯∖u-uλC1Ω¯<β/2 then uλ1<uλ<uλ2 in Ω¯ since Eλ≡E^λ on the set C. Hence uλ is a local minimum for Eλ∖C1(Ω¯). This completes the proof of Lemma 13.


## 4.C1 versus W1,p(x) Local Minimizers of the Energy

The following lemma is crucial in showing multiplicity of solutions. It has been shown in the casep=2 in [18] for the case of critical growth functionals Eλ: H01(Ω)→R, Ω⊂RN, N≥3 and later for critical growth functionals Eλ:W01,p(Ω)→R, 1<p<N, Ω⊂RN, N≥3 in [19]. A key feature of these latter works is the uniform C1,α estimate they obtain for equations like (Pϵ) but involving two p-Laplace operators. Using constraints based on Lp-norms rather than Sobolev norms as in [19], the equations for which uniform estimates are required can be simplified to a standard type involving only one p-Laplace operator. This approach was followed in [20] in the subcritical case, in [21] in the critical case, in [22–24], and also in this work to deal with the boundary value problem involving the nonlinear p(x)-Laplacian case. More precisely, we have the following result.Lemma 14.
Suppose that conditions(H0)–(H5) are satisfied. Let u0∈C1(Ω¯) satisfying(52)u0≥ηdx,∂Ωforsomeη>0be a local minimizer of Eλ in C1Ω¯∩C0Ω¯ topology. Then, u0 is a local minimum of Eλ in W1,pxΩ also.For proving Lemma14, we will need the following uniform L∞-estimates for a family of solutions to Pϵ.Proposition 15.
Letuϵϵ∈(0,1) be a family of solutions to Pϵ, where u0 satisfies (52) and solves Pλ. Let γ>1 be such that (53)supϵ∈0,1⁡fx,uϵLγ+uϵW1,pxΩ<∞.Then, (54)uϵL∞Ω<C.The proof of Proposition15 is a consequence of the results proved in Appendix  A in [13]. Hence, the regularity results of Saoudi and Ghanmi [13] give the following regularity result for weak solutions to problem (Pλ).Theorem 16.
Letu∈W1,pxΩ∩L∞Ω be a solution to problem Pλ. Then, there exists 0<K=KuL∞Ω,p+,p-,δ+,δ-,Ω,N such that any weak solution to problem Pλ belongs to C1,αΩ¯ for some α∈0,1.Proof of Lemma14.
Assume that the conclusion of Lemma14 is not true. We define the following constraint for each ϵ>0:(55)Sϵ≝u∈W1,pxΩ:ρu≝∫Ωuαxαx+1p-∫∂Ωup-≤ϵ.We consider the following constraint minimization problem:(56)Iϵ=infu∈Sϵ⁡Eλu. Firstly, clearly Iϵ>-∞. Moreover, we note that Sϵ is a convex set. Using the trace embeddings we see that Sϵ is also a closed set in W1,pxΩ which implies that Sϵ is weakly closed in W1,pxΩ; with the fact that Eλ is weakly low semicontinuous in W1,pxΩ, it follows that for ϵ∈(0,1)Iϵ is achieved on some uϵ∈Sϵ; that is(57)Eλuϵ=Iϵ,Eλuε<Eλu0∀ϵ∈0,1.Moreover, since Eλuϵ+≤Eλuϵ and uϵ+∈Sϵ, we may assume that uϵ≥0.
We now consider the following two cases.
(1) Letρ(uϵ)<ϵ. Then uϵ is also a local minimizer of Eλ in W1,pxΩ. We now show that Eλ admits Gâteaux derivatives on uϵ to derive that uϵ satisfies the Euler-Lagrange equation associated with Eλ. For this, according to Lemma A.2, in the Appendix, we need to prove that ∃η~>0 such that(58)uϵ≥η~dxdist⁡x,∂Ω,where dx≝dist⁡x,∂Ω. To prove (58), we argue by contradiction: ∀η>0 let(59)Ωη=Supp⁡ηdx-uϵ+,and suppose that Ωη has a nonzero measure.
Letuη=ηdx-uϵ+ and for 0<t≤1 set ξt=Eλuϵ+tuη. Then, there exists c(t) satisfying c(t)>ηt such that inf⁡uϵ+tuη/dx≥c(t)  fort>0. Then, from Lemma A.4  ξ is differentiable for 0<t≤1 and ξ′(t)=Eλ′uϵ+tuη,uη. Thus, (60)ξ′t=∫Ω∇uϵ+tuηpx-2∇uϵ+tuη∇uη+∫Ωuϵ+tuηpx-2uη-∫Ωuϵ+tuη-δxuη-∫Ωfx,uϵ+tuηuη-λ∫Ωuϵ+tuηqx-1uϵ+tuηuη. From(H1)–(H3), we see that (61)ξ′1=∫Ω∇ηdxpx-2∇ηdx∇uη+∫Ωηdxpx-2uη-∫Ωηdx-δxuη-∫Ωfx,ηdxuη-λ∫Ωηdxqx-1ηdxuη<0, for η>0 small enough.
Now, sinces-δ(x)+f(x,s)+λsq(x) is nonincreasing for 0<s small enough uniformly to x∈Ω (by(H1)–(H3)) and from the monotonicity of the operator -Δp(x)u+upx-1u, we have that for 0<η small enough 0≤ξ′1-ξ′t. Therefore, from Taylor’s expansion and since ρuϵ≤ϵ, there exists 0<γ<1 such that(62)0≤Eλuϵ+uη-Eλuϵ=Eλ′uϵ+γuη,uη=ξ′γ.Letting t=γ we have ξ′(γ)≤ξ′(1)<0. We obtain a contradiction with (62) and then uϵ≥ηd(x) for some η>0 (which depends a priori on ϵ). Since uϵ is a local minimizer of Eλ and Eλ is Gâteaux differentiable in uϵ, we get that Eλ′uϵ is defined and Eλ′uϵ=0. Recalling that u_ is the solution to the pure singular problem given by Theorem 4.2 in [16] and from the weak comparison principle, there exist C1>0, C2>0 such that(63)C1dxθ1≤u_≤uϵ,where θ1=maxdx≤σ⁡px/px-1+δx, for some η>0 (independent of ϵ). Since uϵ∈Sϵ and from the fact that uϵ satisfies Pλ, we get that uϵϵ≥0 is uniformly bounded in W1,pxΩ. Now, using Proposition 15 and Theorem 16, we get(64)uϵC1,αΩ¯≤Cfor  someα∈0,1and as ϵ→0+(65)uϵ⟶u0inC1Ω¯ which contradicts the fact that u0 is a local minimizer in C1(Ω¯)∩C0(Ω¯).
Now, we deal with the second case.
(2)ρuϵ=ϵ: we again show that uϵ≥ηd(x) in Ω for some η>0. Taking uη=ηd(x)-uϵ+,  ξt=Eλuϵ+tuη, we obtain as above that ξ′t≤ξ′1<0 for 0<t<1 and 0<η small enough.
Thenξ(t)=Eλuϵ+tuη is decreasing. This implies that Eλuϵ>Eλuϵ+tuη for t>0 and using (52) (66)ρuϵ+tuη<ρuϵ=ϵ. This yields a contradiction with the fact that uϵ is a global minimizer of Eλ on Sϵ. In this case, using Lemma A.4 and from the Lagrange multiplier rule we have(67)Eλ′uϵ=μϵρ′uϵ,forsomeμϵ∈R.
We first show thatμϵ≤0. We argue by contradiction. Suppose that μϵ>0; then there exists φ∈W1,p(x)(Ω) such that (68)Eλ′uϵ,φ<0,ρ′uϵ,φ<0 and then for t small we have(69)Eλuϵ+tφ<Eλuϵ,ρuϵ+tφ<ρuϵ≤ϵ.This contradicts the fact that uϵ is a global minimizer of Eλ in Sϵ.
We deal now with the two following cases.
Case  1 inf ϵ ∈ 0,1 ⁡ μ ϵ ≝ l > - ∞. In this case, we write (67) in its PDE form as(70)Pϵ-Δpxuε+uεpx-1uϵ=1uεδx+fx,uε+μεuεαx-2uε,uε>0inΩ,∇uεpx-2∂uε∂ν=λuεqx+μεuεp--2uε,on⁡∂Ω.
In this case, from (57), we have that supϵ∈(0,1)uϵW1,pxΩ<∞. Hence, we can apply Proposition 15 to conclude that supϵ∈(0,1)uϵL∞Ω<K for some constant K>0 independent of ϵ. Therefore, using Theorem 16 we conclude that uϵC1,αΩ¯≤C for some constant C>0 independent of ϵ and as ϵ→0+(71)uϵ⟶u0inC1Ω¯ which contradicts the fact that u0 is a local minimizer in C1(Ω¯)∩C0(Ω¯).
Now, we deal with the second case.
Case  2 inf ϵ ∈ 0,1 ⁡ μ ϵ = - ∞. From above, we can assume that μϵ≤-1 for 0<ϵ small enough. Furthermore, we can find a number M>0 independent of ϵ>0 and x∈Ω¯, such that 1/sδ(x)+f(x,s)+μεsαx-2s and sqx+μεsp--2ss are negative for all s≥M. Then, from the weak comparison principle (see Lemma 7 and using (uε-M)+ as test function) we have that supϵ∈(0,1)uϵL∞Ω≤M for ϵ>0 small enough. Now, since u0∈W1,p(x)(Ω) is a C1 local minimizer, u0 is a weak solution to Pλ; that is, satisfies ess infK⁡u0>0 over every compact set K⊂Ω and (72)∫Ω∇u0px-2∇u0∇ϕdx+∫Ωu0px-2ϕdx=∫Ω1u0δxϕdx-∫Ωfx,u0ϕdx-λ∫∂Ωu0qx-1u0ϕdx, for all ϕ∈Cc∞(Ω). Therefore, for every function w∈W1,p(x)(Ω), u0 satisfies(73)∫Ω∇u0px-2∇u0∇wdx+∫Ωu0px-2wdx=∫Ω1u0δxwdx-∫Ωfx,u0wdx-λ∫∂Ωu0qx-1u0wdx. Similarly, (74)∫Ω∇uϵpx-2∇uϵ∇wdx+∫Ωuϵpx-2wdx=∫Ω1uϵδxwdx-∫Ωfx,uϵwdx-λ∫∂Ωuϵqx-1uϵwdx. Now, substracting the above relations with w=(uϵ-u0)|uϵ-u0|β-1, with β>1, as a test function in (Pϵ), integrate by parts and use the fact that u↦-Δp(x)u+|u|p(x)-1u is a monotone operator to obtain(75)-με∫Ωuε-u0αxuϵ-u0β-1uε-u0dx+∫∂Ωuε-u0p-+β-1dx≤∫Ω1uεδx-1u0δxuϵ-u0β-1uϵ-u0dx+∫Ωfx,uε-fx,u0uϵ-u0β-1uϵ-u0dx+λ∫∂Ωuεq-1uεu0q-1u0uϵ-u0β-1uϵ-u0dx. Using the bounds of uϵ,u0 we get(76)-με∫Ωuε-u0αxuϵ-u0β-1uε-u0dx+λ∫∂Ωuε-u0p-+β-1dx≤C∫Ωuϵ-u0βdx+λ∫∂Ωuϵ-u0βdx, where C does not depend on β and ϵ. Now, using Hölder’s inequality and the bounds of uϵ combined with Lemma 4 we obtain (77)∫Ωuεβ≤CuεLαx+β-1β≤∫Ωuεαx+β-1β/α-+β-1. Therefore(78)-με∫Ωuε-u0αxuϵ-u0β-1uε-u0dx+λ∫∂Ωuε-u0p-+β-1dx≤CΩ∫Ωuϵ-u0αx+β-1dx+λ∫∂Ωuϵ-u0αx+β-1dxβ/α-+β-1. Thus for any β>1(79)-μϵuϵ-u0Lαx+β-1Ωα--1+uϵ-u0Lαx+β-1∂Ωα--1≤CΩ.Passing to the limit in (79) β→+∞ we get(80)sup⁡-μϵuϵ-u0L∞Ωα--1+uϵ-u0L∞∂Ωα--1≤CΩ.Then, using (80) combined with Proposition 15, the uniform L∞ bounds for uεε∈(0,1) in Ω as well as ∂Ω, we get that the right-hand side terms in (Pε) are uniformly bounded in L∞(Ω) and in L∞(∂Ω) from which as in the first case we obtain that uϵ(0<ϵ≤1) is bounded in C1,α(Ω¯) independently of ϵ. Finally, using Ascoli-Arzela Theorem we find a sequence ϵn→0+ such that (81)uϵn⟶u0inC1Ω¯. It follows that, for ϵ>0 sufficiently small,(82)Eλuϵn<Eλu0, which contradicts the fact that u0 is a local minimizer of Eλ for the C1(Ω¯)∩C0(Ω¯) topology. The proof of Lemma 14 is now completed.


## 5. Existence of a Second Weak Solution for0<λ<Λ

In this section, we fixλ∈(0,Λ) and let 0<λ1<λ<λ2<Λ, uλ1 and uλ2 be as in Section 3. Now, we are able to show the existence of a second solution using the generalized Mountain Pass Theorem. Since the functional is not C1, we use the cut-off functional E¯λ defined in (86). Define the cut-off functionsg ¯ : Ω × R → R by (83)g¯x,s=1sδx,ifs>uλx,1uxλδx,ifs≤uλx,f ¯ : Ω × R → R by (84)f¯x,s=fx,s,ifs>uλx,fx,uλx,ifs≤uλx,h ¯ : ∂ Ω × R → R by (85)h¯x,s=ψxsqx,ifs>uλx,ψxuλqxx,ifs≤uλx, and define now the corresponding functional E¯λ:W1,p(x)(Ω)→R given by(86)E¯λu=∫Ω∇upxpx+∫Ωupxpx-∫ΩG¯x,udx-∫ΩF¯x,udx-λ∫∂ΩH¯x,udx,where G¯(x,u)=∫0u‍g¯(x,t)dt,F¯(x,u)=∫0u‍f¯(x,t)dt, and H¯(x,u)=∫0u‍h¯(x,t)dt. Firstly, we prove the following lemma on compactness of Palais-Smale sequences.Lemma 17.
The functionalE¯λ satisfies the Palais-Smale condition.Proof.
Let{un} be a (PS) sequence; namely, E¯λ(un) is bounded and E-λ′(un)→0 when n→∞. Then,(87)c≥Eλun-1θEλ′un,un+1θEλ′un,un≥∫Ω∇unpxpx+∫Ωunpxpx-∫ΩG¯x,un-∫ΩF¯x,un-1θ∫Ω∇unpxpx+∫Ωunpxpx-∫ΩG¯x,un-∫ΩF¯x,un-λ∫∂ΩH¯x,un+1θEλ′un,un.Now, we estimate the boundary term from above as follows:(88)∫∂ΩH¯x,un=∫∂Ω∩un≤1H¯x,un+∫∂Ω∩un>1H¯x,un≤K1+∫∂Ω∩un≤1unqx+1qx+1≤K1+1q-+1∫∂Ωunq++1≤K1+K2∫Ω∇unp-q++1/p-.Hence, taking (88) in (87) and using(H2) combined with (7) we get(89)≥1p+-1θ∫Ω∇unpx+unpx-K1∫Ω∇unp-q++1/p-+1θE-λ′un,un-K2≥1p+-1θ∫Ω∇unpx+unpx-K1∫Ω∇unp-q++1/p--1θE-λ′ununW1,pxΩ-K2.Now, using Lemma 4 and the fact that Lp(x)⊂Lp-(Ω), we get(90)∇unLpxq++1≥1p+-1θ∫Ω∇unpx+unpx-1θE-λ′ununW1,pxΩ-K2≥1p+-1θ∇unLpxp-+unLpxp--1θ∇unLpx+unLpx-K2.Hence, unW1,p(x)(Ω) is bounded. Without loss of generality, we assume that there exists a subsequence of {un} such that un⇀u0. Therefore, using Theorems 5 and 6 we get(91)limn→∞∫Ωg¯x,unun-u0⟶0asn⟶∞,limn→∞∫Ωf¯x,unun-u0⟶0asn⟶∞,limn→∞∫∂Ωh¯x,unun-u0⟶0asn⟶∞.Observe that(92)E¯λun,un-u0=∫Ω∇unpx-2un∇un-u0+unpx-2unun-u0-∫Ωg¯x,unun-u0-∫Ωf¯x,un-∫∂Ωh¯x,unun-u0.We already know that(93)E¯λun,un-u0⟶0asn⟶∞.Using (91), we obtain (94)∫Ω∇unpx-2∇un∇un-u0+unpx-2unun-u0⟶0asn⟶∞. This together with the convergence of un⇀u0 in W1,p(x)(Ω) implies that un→u0 strongly in W1,pxΩ; that is, E¯λ satisfies the (PS) condition. The proof of Lemma 17 is now completed.Proof of Theorem3.
Firstly, note thatE¯λ(u)≥E¯λ(uλ) for any solution u of (50). Hence, as in Section 3 we can conclude that uλ is a local minimum for E¯λ in W1,p(x)(Ω). By the strong comparison principle and Hopf lemma, we can conclude that any critical point of E¯λ is also a critical point of Eλ and hence u also solves (Pλ). It is easy to see that uλ1 and uλ2 are a subsolution and a supersolution to the problem associated with the functional energy E¯λ. Therefore, using the approach as in Theorem 2, we prove that this problem has a solution vλ∈[uλ1,uλ2]∩C1(Ω¯) such that vλ is a local minimizer of E¯λ in the C1 topology. Now, by the comparison principle we can see that vλ≥uλ and also vλ solves (Pλ). If vλ≢uλ the conclusion of Theorem 3 holds. That is, we can assume vλ≡uλ and uλ is a strict local minimum of E¯λ in the W1,p(x)(Ω) topology. Then, from Lemma A.4, the functional E¯λ∈C1(W1,p(x)(Ω),R) and note that E¯λ(tφ)→-∞ as t→∞. Thus, we can apply Lemma 17 combined with the Mountain Pass Theorem to conclude that problem (Pλ) has a solution vλ such that vλ≠uλ. Therefore the proof of Theorem 3 is now completed.



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