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Advances in Nonlinear Analysis

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On the existence of a weak solution for some singular p(x)-biharmonic equation with Navier boundary conditions

Khaled Kefi
  • Corresponding author
  • Community College of Rafha, Northern Border University, 73222 Arar, Kingdom of Saudi Arabia; and Mathematics Department, Faculty of Sciences Tunis El Manar, 1060 Tunis, Tunisia
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/ Kamel Saoudi
Published Online: 2018-03-13 | DOI: https://doi.org/10.1515/anona-2016-0260

Abstract

In the present paper, we investigate the existence of solutions for the following inhomogeneous singular equation involving the p(x)-biharmonic operator:

{Δ(|Δu|p(x)-2Δu)=g(x)u-γ(x)λf(x,u)in Ω,Δu=u=0on Ω,

where ΩN (N3) is a bounded domain with C2 boundary, λ is a positive parameter, γ:Ω¯(0,1) is a continuous function, pC(Ω¯) with 1<p-:=infxΩp(x)p+:=supxΩp(x)<N2, as usual, p*(x)=Np(x)N-2p(x),

gLp*(x)p*(x)+γ(x)-1(Ω),

and f(x,u) is assumed to satisfy assumptions (f1)–(f6) in Section 3. In the proofs of our results, we use variational techniques and monotonicity arguments combined with the theory of the generalized Lebesgue Sobolev spaces. In addition, an example to illustrate our result is given.

Keywords: Navier boundary condition; singular problem; variational methods; existence results; generalized Lebesgue Sobolev spaces

MSC 2010: 35J20; 35J60; 35G30; 35J35

1 Introduction

Let Ω be a smooth bounded domain in N (N3) with C2 boundary condition. In this paper, we are dealing with the following problem:

{Δ(|Δu|p(x)-2Δu)=g(x)u-γ(x)λf(x,u)in Ω,Δu=u=0on Ω,(P-+λ)

where λ is a positive parameter, γ(x)C(Ω¯) satisfying 0<γ-=infxΩγ(x)γ+=supxΩγ(x)<1, pC(Ω¯) with 1<p-:=infxΩp(x)p+:=supxΩp(x)<N2, as usual, p*(x)=Np(x)N-2p(x), and

gLp*(x)p*(x)+γ(x)-1(Ω)

and is almost everywhere positive in Ω. In the sequel, X will denote the Sobolev space W2,p(x)(Ω)W01,p(x)(Ω). Associated to problem (P-+λ), we have the singular functional Iλ:X given by

Iλ(u)=J(u)-Φλ(u),

where

J(u)=Ω1p(x)|Δu|p(x)dx

and

Φλ(u)=Ωg(x)1-γ(x)|u|1-γ(x)dxλΩF(x,u(x))dx,

where F(x,t)=0tf(x,s)ds.

Definition 1.1.

If for all vX,

Ω|Δu|p(x)-2ΔuΔvdx=Ωg(x)|u|-γ(x)vdxλΩf(x,u)vdx,

then uX is called a weak solution of (P-+λ).

The operator Δp(x)2u:=Δ(|Δu|p(x)-2Δu) is called the p(x)-biharmonic operator of fourth order, where p is a continuous non-constant function. This differential operator is a natural generalization of the p-biharmonic operator Δp2u:=Δ(|Δu|p-2Δu), where p>1 is a real constant. However, the p(x)-biharmonic operator possesses more complicated non-linearity than the p-biharmonic operator, due to the fact that Δp(x)2 is not homogeneous. This fact implies some difficulties; for example, we can not use the Lagrange multiplier theorem in many problems involving this operator.

The study of this kind of operators occurs in interesting areas such as electrorheological fluids (see [19]), elastic mechanics (see [25]), stationary thermo-rheological viscous flows of non-Newtonian fluids, image processing (see [6]) and the mathematical description of the processes filtration of an ideal barotropic gas through a porous medium (see [1]).

Problem (P-+λ) is a new variant of p(x)-biharmonic equations due to the singular term and the indefinite one. Note that results for p(x)-Laplace equations with singular non-linearity are rare. Meanwhile, elliptic and singular elliptic equations involving the p(x)-Laplace and the p(x)-biharmonic operators can be found in [1, 2, 5, 8, 9, 10, 13, 14, 17, 18, 21, 20, 22, 23].

Recently, Ayoujil and Amrouss [4] studied the following problem:

{Δ(|Δu|p(x)-2Δu)=λ|u|q(x)-2uin Ω,Δu=u=0on Ω.(P)

In the case when maxxΩq(x)<minxΩp(x), they proved that the energy functional associated to problem (P) has a nontrivial minimum for any positive λ; see [4, Theorem 3.1]. In the case when minxΩq(x)<minxΩp(x) and q(x) has a subcritical growth, they used Ekeland’s variational principle in order to prove the existence of a continuous family of eigenvalues which lies in a neighborhood of the origin. Finally, when

maxxΩp(x)<minxΩq(x)maxxΩq(x)<Np(x)N-2p(x),

they showed (see [4, Theorem 3.8]) that for every Λ>0 the energy functional Φλ corresponding to (P) has a Mountain Pass-type critical point which is nontrivial and nonnegative, and hence Λ=(0,+), where Λ is the set of the eigenvalues. The same problem, for p(x)=q(x), is studied by Ayoujil and Amrouss in [3]. They established the existence of infinitely many eigenvalues for problem (P) by using an argument based on the Ljusternik–Schnirelmann critical point theory. They showed that supΛ=+, and they pointed out that only under special conditions, which are somehow connected with a kind of monotony of the function p(x), one has infΛ>0 (this is in contrast with the case when p(x) is a constant, where one always has infΛ>0).

Later, Ge, Zhou and Wu [10] considered the following problem:

{Δ(|Δu|p(x)-2Δu)=λV(x)|u|q(x)-2uin Ω,Δu=u=0on Ω,(P1)

where V is an indefinite weight and λ is a positive real number. They considered different situations concerning the growth rates, and they proved, using the mountain pass lemma and Ekeland’s principle, the existence of a continuous family of eigenvalues. A recent paper concerning this type of problems is [12].

Inspired by the above-mentioned papers, we study problem (P-+λ), which contains a singular term and indefinite many more general terms than the one studied in [10]. In this new situation, we will show the existence of a weak solution for problem (P-+λ). The paper is organized as follows: In Section 2, we recall some definitions concerning variable exponent Lebesgue spaces, Lp(x)(Ω), as well as Sobolev spaces, Wk,p(x)(Ω). Moreover, some properties of these spaces will also be exhibited to be used later. Our main results are stated in Section 3. The proofs of our results will be presented in Section 4 and Section 5.

2 Notations and preliminaries

To study p(x)-biharmonic problems, we need some results on the spaces Lp(x)(Ω), W1,p(x)(Ω) and Wk,p(x)(Ω) (for details, see [18, 9]) and some properties of the p(x)-biharmonic operator, which will be needed later. Set

C+(Ω¯):={h:hC(Ω¯),h(x)>1 for all xΩ¯}.

Let p be a Lipschitz continuous function on Ω¯. We set 1<p-:=minxΩ¯p(x)p+=maxxΩ¯p(x)< and

Lp(x)(Ω)={u:Ω measurable such that Ω|u(x)|p(x)dx<}.

We recall the following so-called Luxemburg norm on this space defined by the formula

|u|p(x)=inf{μ>0:Ω|u(x)μ|p(x)dx1}.

Clearly, when p(x)=p, a positive constant, the space Lp(x)(Ω) reduces to the classical Lebesgue space Lp(Ω), and the norm |u|p(x) reduces to the standard norm

uLp=(Ω|u|pdx)1pin Lp(Ω).

For any positive integer k, as in the constant exponent case, let

Wk,p(x)(Ω)={uLp(x)(Ω):DαuLp(x)(Ω),|α|k},

where α=(α1,,αN) is a multi-index, |α|=i=1Nαi and

Dαu=|α|uα1x1αNxN.

Then Wk,p(x)(Ω) is a separable and reflexive Banach space equipped with the norm

uk,p(x)=|α|k|Dαu|p(x).

Furthermore, W0k,p(x)(Ω) is the closure of C0(Ω) in Wk,p(x)(Ω). Let Lp(x)(Ω) be the conjugate space of Lp(x)(Ω) with 1p+1p=1. Then the Hölder-type inequality

|Ωuvdx|(1p-+1(p)-)|u|p(x)|v|p(x),uLp(x)(Ω),vLp(x)(Ω)(2.1)

holds.

The modular on the space Lp(x)(Ω) is the map ρp(x):Lp(x)(Ω) defined by

ρp(x)(u):=Ω|u|p(x)dx,

and it satisfies the following propositions.

Proposition 2.1 (see [16]).

For all uLp(x)(Ω), we have the following assertions:

  • (i)

    |u|p(x)<1 (resp. =1, >1 ) if and only if ρp(x)(u)<1 (resp. =1, >1 ).

  • (ii)

    min(|u|p(x)p-,|u|p(x)p+)ρp(x)(u)max(|u|p(x)p-,|u|p(x)p+).

  • (iii)

    ρp(x)(un-u)0 if and only if |un-u|p(x)0.

Proposition 2.2 (see [7]).

Let p and q be two measurable functions such that pL(Ω) and 1p(x)q(x) for a.e. xΩ. Let uLq(x)(Ω), u0. Then

min(|u|p(x)q(x)p+,|u|p(x)q(x)p-)||u|p(x)|q(x)max(|u|p(x)q(x)p-,|u|p(x)q(x)p+).

For more details concerning the modular, see [9, 16].

Definition 2.3.

Assuming that E and F are Banach spaces, we define the norm on the space X:=EF as uX=uE+uF .

In order to discuss problems (P-+λ), we need some theories on the space X:=W01,p(x)(Ω)W2,p(x)(Ω). From Definition 2.3 we know that for any uX,

u=u1,p(x)+u2,p(x),

and thus

u=|u|p(x)+|u|p(x)+|α|=2|Dαu|p(x).

Zang and Fu [24], proved the equivalence of the norms, and they even proved that the norm |Δu|p(x) is equivalent to the norm u (see [24, Theorem 4.4]). Let us choose on X the norm defined by u=|Δu|p(x). Note that (X,) is also a separable and reflexive Banach space and that the modular is defined as ρp(x):X by ρp(x)(Δu)=Ω|Δu|dx and satisfies the same properties as in Proposition 2.1. Hereafter, let

p*(x)={Np(x)N-2p(x),p(x)<N2,+,p(x)N2.

Remark 2.4.

If qC+(Ω¯) and q(x)<p*(x) for any xΩ, by [4, Theorem 3.2] we deduce that X is continuously and compactly embedded in Lq(x)(Ω).

Throughout this paper, the letters k,c,C,Ci,i=1,2, , denote positive constants which may change from line to line.

3 Hypotheses and main results

Let us impose the following hypotheses on the non-linearity f:Ω¯×:

  • (f1)

    f is a C1 function such that f(x,0)=0.

  • (f2)

    There exists Ω1Ω with |Ω1|>0, and a nonnegative function h1 on Ω1 such that h1Ls1(x)(Ω) with

    lim|t|0f(x,t)h1(x)|t|r1(x)-1=0for xΩ uniformly.

  • (f3)

    There exists a positive function h on Ω such that hLs(x)(Ω) and

    lim|t|+f(x,t)h(x)|t|r(x)-1=0for xΩ uniformly,

    where s,s1,r and r1C(Ω¯) are such that 1<max{r(x),r1(x)}<p(x)<N2<min(s(x),s1(x)) for all xΩ.

  • (f4)

    There exists A>0 such that

    ΩF(x,t)dx>0for all t>A.

  • (f5)

    f(x,t)Ch(x)|t|r(x)-2t for all t and all xΩ¯, where C is a positive constant, hLs(x)(Ω) and s,rC(Ω¯) are such that for all xΩ¯ we have 1<r(x)<p(x)<N2<s(x).

  • (f6)

    There exists Ω1Ω with |Ω1|>0 such that f(x,t),h(x)>0 in Ω1.

Some remarks regarding the hypotheses are in order.

Remark 3.1.

Under assumptions (f3) and (f4), we have the following assertions.

  • (i)

    (1-γ(x))p(x)<p(x)<p*(x) for any xΩ¯, so the injection of XL(1-γ(x))p(x)(Ω) is compact and continuous.

  • (ii)

    s(x)r(x)<p*(x) for any xΩ¯, where 1s(x)+1s(x)=1, so XLs(x)r(x)(Ω) is compact and continuous.

  • (iii)

    s1(x)r1(x)<p*(x) for any xΩ¯, where 1s1(x)+1s1(x)=1, so XLs1(x)r1(x)(Ω) is compact and continuous.

  • (iv)

    XLp*(x)(Ω) is continuous.

Moreover, under conditions (f5) and (f6), we remark the following.

Remark 3.2.

  • (i)

    There exists K>0 such that ηf(x,η)Kr(x)F(x,η) for all xΩ1,η+.

  • (ii)

    Due to condition (f5), there exists C>0 such that

    F(x,η)Ch(x)|η|r(x)-1ηin Ω×.

  • (iii)

    Conditions (f5) and (f6) assure that F(x,η)>0 for all xΩ1,η.

  • (iv)

    Put f(x,t)=Ch(x)|t|r(x)-2t,xΩ,t. Then the first condition in the remark is satisfied.

Here we state our main results asserted in the following two theorems.

Theorem 3.3.

Assume that hypotheses (f1), (f2), (f3) and (f4) are fulfilled. Then for all λ>0 problem (P-λ) has at least one nontrivial weak solution with negative energy.

Theorem 3.4.

Assume that hypotheses (f5) and (f6) are fulfilled. Then for all λ>0 problem (P+λ) has at least one nontrivial weak solution with negative energy.

4 Proof of Theorem 3.3

The study of the existence of solutions to problem (P-λ) is done by looking for critical points to the functional I-λ:X defined by

I-λ(uλ)=Ω|Δuλ|p(x)p(x)dx-Ωg(x)uλ1-γ(x)1-γ(x)dx+λΩF(x,uλ)dx

in the Sobolev space X. The proof of the Theorem 3.3 is organized in several lemmas. Firstly, under Remark 3.1 one has

|uλ|(1-γ(x))p(x)C2uλfor all uλX(4.1)

and

|uλ|s(x)r(x)C3uλfor all uλX.

Now, we are in a position to show that I-λ possesses a nontrivial global minimum point in X.

Lemma 4.1.

Under assumptions (f2), (f3) and (f4), the functional I-λ is coercive on X.

Proof.

First, we recall that in view of assumptions (f3), (f4), inequality (4.1), Remark 3.1 and Proposition 2.1 one has for any uλX with uλ>max(1,A),

I-λ(uλ)Ω1p(x)|Δuλ|p(x)dx-C4|g|p*(x)p*(x)+γ(x)-1||uλ|1-γ(x)|p*(x)+λΩF(x,uλ)dxΩ1p(x)|Δuλ|p(x)dx-C4|g|p*(x)p*(x)+γ(x)-1||uλ|1-γ(x)|p(x)Ω1p(x)|Δuλ|p(x)dx-C4|g|p*(x)p*(x)+γ(x)-1uλ1-γ-1p+ρp(x)(Δuλ)-C4|g|p*(x)p*(x)+γ(x)-1uλ1-γ-1p+uλp--C4|g|p*(x)p*(x)+γ(x)-1uλ1-γ-.

Since 1-γ-<p-, we infer that I-λ(uλ) as uλ; in other words, I-λ is coercive on X. The proof of Lemma 4.1 is now completed. ∎

Lemma 4.2.

Suppose assumptions (f2) and (f3) are fulfilled. Then there exists φX such that φ0, φ0 and I-λ(tφ)<0 for t>0 small enough.

Proof.

Let φC0(Ω) such that supp(φ)Ω1Ω,φ=1 in a subset Ωsupp(φ) and 0φ1 in Ω1. Using assertions on the functions g and F and assumption (f2), we have

I-λ(tφ)=Ω1p(x)|Δtφ|p(x)dx-Ω11-γ(x)g(x)|tφ|1-γ(x)dx+λΩF(x,tφ)dxtp-p-ρp(x)(Δφ)-t1-γ-1-γ-Ωg(x)|φ|1-γ(x)dx+λC1tr1-Ω1h1(x)|φ|r1(x)dxtr1-[1p-ρp(x)(Δφ)+λC1Ω1h1(x)|φ|r1(x)dx]-t1-γ-1-γ-Ωg(x)|φ|1-γ(x)dx,

so

I-λ(tφ)<0for t<ψ1r1--(1-γ-)

with

0<ψ<min{1,11-γ-Ωg(x)|φ|1-γ(x)dx1p-ρp(x)(Δφ)+λC1Ω1h1(x)|φ|r1(x)dx}.

Finally, we point out, using the hypothesis on φ and the definition of the modular on X, that

1p-ρp(x)(Δφ)+λCΩ1h1(x)|φ|r1(x)dx>0.

In fact, if

1p-ρp(x)(Δφ)+λC1Ω1h1(x)|φ|r1(x)dx=0,

then ρp(x)(Δφ)=0, and consequently φ=0, which contradicts the choice of φ and gives the proof of Lemma 4.2. ∎

In the sequel, we put mλ=infuλXI-λ(uλ). Then we have the following lemma.

Lemma 4.3.

Let λ0, γC(Ω¯,(0,1)),

gLp*(x)p*(x)+γ(x)-1(Ω)

with g(x)>0 for almost every xΩ, and assume that hypothesis (f1), (f2), (f3) and (f4) are fulfilled. Then I-λ reaches its global minimizer in X, that is, there exists uλX such that I-λ(uλ)=mλ<0.

Proof.

Let {un} be a minimizing sequence, that is to say I-λ(un)mλ. Suppose {un} is not bounded, so un+ as n+. Since Iλ is coercive, we have

I-λ(un)+as un+.

This contradicts the fact that {un} is a minimizing sequence, so {un} is bounded in X, and therefore, up to a subsequence, there exists uλX such that

unuλweakly in X,unuλstrongly in Ls(x)(Ω), 1s(x)<p*(x),un(x)uλ(x)a.e. in Ω.

Since J:X is sequentially weakly lower semi-continuous (see [12]), we have

Ω1p(x)|Δuλ|p(x)lim infn+Ω1p(x)|Δun|p(x).(4.2)

On the other hand, by Vital’s theorem (see [16, p. 113]), we can claim that

limnΩg(x)|un|1-γ(x)dx=Ωg(x)|uλ|1-γ(x)dx.(4.3)

Indeed, we only need to prove that

{Ωg(x)|un|1-γ(x)dx:n}

is equi-absolutely-continuous. Note that {un} is bounded in X, so Remark 3.1 implies that {un} is bounded in Lp*(x)(Ω). For every ε>0, using Proposition 2.1 and the absolutely-continuity of

Ω|g(x)|p*(x)p*(x)+γ(x)-1dx,

there exist ζ,ξ>0 such that

|g|p*(x)p*(x)+γ(x)-1ζΩ|g(x)|p*(x)p*(x)+γ(x)-1dxεζfor any Ω2Ω with |Ω2|<ξ.

Consequently, by the Hölder inequality and Proposition 2.1 one has

Ω|g(x)||un|1-γ(x)dx|g|p*(x)p*(x)+γ(x)-1||un|1-γ(x)|p*(x)|g|p*(x)p*(x)+γ(x)-1|un|(1-γ(x))p*(x)k.

Since (1-γ(x))p*(x)<p*(x), we have

|un|(1-γ(x))p*(x)C7|un|p*(x),

so

Ω|g(x)||un|1-γ(x)dx|g|p*(x)p*(x)+γ(x)-1||un|1-γ(x)|p*(x)|g|p*(x)p*(x)+γ(x)-1|un|(1-γ(x))p*(x)k<εC7k|un|p*(x)k.

Since |un|p*(x) is bounded, claim (4.3) is valid.

In what follows, we remark, using assumptions (f2) and (f3), that for all ε>0 there exists Cε such that

|F(x,uλ(x))|εC1r1-|h1(x)||uλ|r1(x)+CεCr-|h(x)||uλ|r(x).

Then by the Hölder inequality one has

Ω|F(x,uλ(x)|εC1r1-|h1|s1(x)||uλ|r1(x)|s1(x)+CεCr-|h|s(x)||uλ|r(x)|s(x).

Besides, if unuλ in X, then we have strong convergence in Ls(x)r(x)(Ω) and Ls1(x)r1(x)(Ω). So the Lebesgue dominated convergence theorem and Proposition 2.2 enable us to state the following assertion: If

uλΩF(x,uλ(x))dx

is weakly continuous, then

limn+Φ-λ(un)=Φ-λ(uλ).(4.4)

Using (4.2), (4.3) and (4.4), we deduce that I-λ is weakly lower semi-continuous, and consequently

mλI-λ(uλ)lim infn+I-λ(un)=mλ.

The proof of Lemma 4.3 is now completed. ∎

Proof of Theorem 3.3.

Now, let us show that the weak limit uλ is a weak solution of problem (P-λ) if λ>0 is sufficiently large. Firstly, observe that I-λ(0)=0. So, in order to prove that the solution is nontrivial, it suffices to prove that there exists λ>0 such that

infuλXI-λ(uλ)<0for all λ>0.

For this purpose, we consider the variational problem with constraints

λ:=inf{Ω1p(x)|Δw|p(x)dx+Ωg(x)1-γ(x)|w|1-γ(x)dx:wX and ΩF(x,w(x))dx=1},(4.5)

and define

Λ*:=inf{λ>0:(P-λ) admits a nontrivial weak solution}.

From above we have

I-λ(uλ)=λ-λ<0for any λ>λ.

Therefore, the above remarks show that λΛ* and that problem (P-λ) has a solution for all λ>λ.

We now argue that problem (P-λ) has a solution for all λ>Λ*. Fixing λ>Λ*, by the definition of Λ* we can take μ(Λ*,λ) such that I-μ has a nontrivial critical point uμX. Since μ<λ, we obtain that uμ is a sub-solution of problem (P-λ). We now want to construct a super-solution of problem (P-λ) which dominates uμ. For this purpose, we introduce the constrained minimization problem

inf{I-λ(w):wX and wuμ}.

By using the previous arguments to treat (4.5), follows that the above minimization problem has a solution uλ>uμ. Moreover, uλ is also a weak solution of problem (P-λ) for all λ>Λ*. With the arguments developed in [15], we deduce that problem (P-λ) has a solution if λ=Λ*.

Now, it remains to show that Δuλ=0 on Ω. Due to the above arguments, one has

Ω|Δuλ|p(x)-2ΔuλΔvdx=Ωm(x)vdxfor all vX,(4.6)

where

m(x)=g(x)uλ-γ(x)-λf(x,uλ).

Relation (4.6) implies that

Ω|Δuλ|p(x)-2ΔuλΔvdx=Ωm(x)vdxfor all vC0(Ω).(4.7)

Let ζ be the unique solution of the problem

{Δζ=m(x)in Ω,ζ=0on Ω.

Relation (4.7) yields

Ω|Δuλ|p(x)-2ΔuλΔvdx=Ω(Δζ)vdxfor all vC0(Ω).

Using the Green formula, we have

Ω(Δζ)vdx=ΩζΔvdx.

Therefore,

Ω|Δuλ|p(x)-2ΔuλΔvdx=ΩζΔvdxfor all vC0(Ω).(4.8)

On the other hand, for all uλ~C0(Ω) there exists a unique vC0(Ω) such that Δv=uλ~ in Ω. Thus, relation (4.8) can be rewritten as

Ω(|Δuλ|p(x)-2Δuλ-ζ)uλ~dx=0for all uλ~C0(Ω).

Applying the fundamental lemma of the calculus of variations, we deduce that

|Δuλ|p(x)-2Δuλ-ζ=0in Ω.

Since ζ=0 on Ω, we conclude that Δuλ=0 on Ω. Thus, uλ is a nontrivial weak solution of problem (P-λ) such that Δuλ=0. This completes the proof of Theorem 3.3. ∎

5 Proof of Theorem 3.4

The proof of Theorem 3.4 is organized in several lemmas. Firstly, we show the existence of a local minimum for I+λ in a small neighborhood of the origin in X.

Lemma 5.1.

Under assumption (f5), the functional I+λ is coercive on X.

Proof.

Using Remark 3.1, inequality (2.1) and Proposition 2.1, we obtain that for any vλX with vλ>1,

I+λ(vλ)=Ω1p(x)|Δvλ|p(x)dx-11-γ(x)Ωg(x)|vλ|1-γ(x)dx-λΩF(x,vλ(x))dx1p+ρp(x)(Δvλ)-11-γ+Ωg(x)|vλ|1-γ(x)dx-λΩF(x,vλ(x))dx1p+vλp--11-γ+|g|p*(x)p*(x)+γ(x)-1||vλ|1-γ(x)|p*(x)-CλΩh(x)|vλ|r(x)dx1p+vλp--11-γ+|g|p*(x)p*(x)+γ(x)-1||vλ|1-γ(x)|p(x)-Cλ|h|s(x)||vλ|r(x)|s(x)dx1p+vλp--11-γ+|g|p*(x)p*(x)+γ(x)-1min(|vλ|(1-γ(x))p(x)1-γ+,|vλ|(1-γ(x))p(x)1-γ-)-Cλ|h|s(x)min(|vλ|s(x)r(x)r+,|vλ|s(x)r(x)r-)1p+vλp--11-γ+|g|p*(x)p*(x)+γ-1min(C11-γ+vλ1-γ+,C11-γ-vλ1-γ-)-Cλ|h|s(x)min(C2r+vλr+,C2r-vλr-).

Since 1-γ-<r+<p-, we infer that Iλ(vλ) as vλ and I+λ is coercive on X. This ends the proof of Lemma 5.1. ∎

Lemma 5.2.

Under assumptions (f5) and (f6), there exists φX such that φ0,φ0 and I+λ(tφ)<0 for t>0 small enough.

Proof.

Let φC0(Ω) such that supp(φ)Ω1Ω,φ=1 in a subset Ωsupp(φ) and 0φ1 in Ω1. Using assertions on the functions g and F, we have

I+λ(tφ)=Ω1p(x)|Δtφ|p(x)dx-Ω11-γ(x)g(x)|tφ|1-γ(x)dx-λΩF(x,tφ)dxΩtp(x)p-|Δφ|p(x)dx-Ωg(x)1-γ(x)|tφ|1-γ(x)dx-λΩF(x,tφ)dxtp-p-Ω|Δφ|p(x)dx-t1-γ-Ωg(x)|φ|1-γ(x)1-γ(x)dx-λΩ1F(x,tφ)dx.

Then

I+λ(tφ)tp-p-Ω|Δφ|p(x)dx-t1-γ-1-γ-Ωg(x)|φ|1-γ(x)dx.

Since p->1-γ-, we have I+λ(tφ)<0 for t<ψ1/(p--(1-γ-)) with

0<ψ<min{1,p-1-γ-Ωg(x)|φ|1-γ(x)dxρp(x)(Δφ)}.

Finally, we point out that ρp(x)(Δφ)>0. In fact, if ρp(x)(Δφ)=0, then φ=0, and consequently φ=0 in Ω, which is a contradiction. ∎

In the sequel, put mλ1=infvλXI+λ(vλ). As a last proposition, we have the following.

Lemma 5.3.

Let λ0, γC(Ω¯,(0,1)),

gLp*(x)p*(x)+γ(x)-1(Ω)

with g(x)>0 for almost every xΩ, and assume that assertions (f5) and (f6) hold. Then I+λ reaches its global minimizer in X, that is, there exists vλX such that I+λ(vλ)=mλ1<0.

Proof.

The proof of Lemma 5.3 is word for word as the one of Lemma 4.3. ∎

Proof of Theorem 3.4.

From Lemma 5.3, vλ is a local minimizer for I+λ, with I+λ(vλ)=mλ<0, which implies that vλ is nontrivial. Now, we prove that vλ is a positive solution of problem (P+λ). Our proof is inspired by Saoudi and Ghanmi in [11].

Let ϕX and 0<ϵ<1. We define ΨX by Ψ:=(vλ+ϵϕ)+, where (vλ+ϵϕ)+=max{vλ+ϵϕ,0}. Since vλ is a local minimizer for I+λ, one has

0Ω|Δvλ|p(x)-2ΔvλΔΨdx-Ωg(x)vλ-γ(x)Ψdx-λΩf(x,vλ)Ψdx={x:vλ+ϵϕ>0}|Δvλ|p(x)-2ΔvλΔ(vλ+ϵϕ)dx-{x:vλ+ϵϕ>0}g(x)u-γ(x)(vλ+ϵϕ)dx-λ{x:vλ+ϵϕ>0}f(x,vλ)(vλ+ϵϕ)dx=Ω|Δvλ|p(x)-2ΔvλΔ(vλ+ϵϕ)dx-Ωg(x)u-γ(x)(vλ+ϵϕ)dx-λΩf(x,vλ)(vλ+ϵϕ)dx-{x:vλ+ϵϕ0}|Δvλ|p(x)-2ΔvλΔ(vλ+ϵϕ)dx-{x:vλ+ϵϕ0}g(x)u-γ(x)(vλ+ϵϕ)dx-λ{x:vλ+ϵϕ0}f(x,vλ)(vλ+ϵϕ)dx=Ω|Δvλ|p(x)dx-Ωg(x)vλ1-γ(x)dx-λΩf(x,vλ)vλdx+ϵ(Ω|Δvλ|p(x)-2Δvλvλϕdx-Ωg(x)vλ-γ(x)ϕdx-λΩf(x,vλ)ϕdx)+ϵ2Ω|Δvλ|p(x)-2ΔvλΔϕdx-{x:vλ+ϵϕ0}|Δvλ|p(x)-2ΔvλΔ(vλ+ϵϕ)dx-{x:vλ+ϵϕ0}g(x)vλ-γ(x)(vλ+ϵϕ)dx-λ{x:vλ+ϵϕ0}f(x,vλ)(vλ+ϵϕ)dx=ϵ(Ω|Δvλ|p(x)-2Δvλuϕdx-Ωg(x)vλ-γ(x)ϕdx-λΩf(x,vλ)ϕdx)+ϵ2Ω|Δvλ|p(x)-2ΔvλΔϕdx-{x:vλ+ϵϕ0}|Δvλ|p(x)-2ΔvλΔ(vλ+ϵϕ)dx-{x:vλ+ϵϕ0}g(x)vλ-γ(x)(vλ+ϵϕ)dx-λ{x:vλ+ϵϕ0}f(x,vλ)(vλ+ϵϕ)dxϵ(ϵΩ|Δvλ|p(x)-2ΔvλΔϕdx-Ωg(x)vλ-γ(x)ϕdx-λΩf(x,vλ)ϕdx)+ϵΩ|Δvλ|p(x)-2Δvλvλϕdx-ϵ2{x:vλ+ϵϕ0}|Δvλ|p(x)-2ΔvλΔϕdxϵ(Ω|Δvλ|p(x)-2ΔvλΔϕdx-Ωg(x)vλ-γ(x)ϕdx-λΩf(x,vλ)ϕdx)-ϵ{x:vλ+ϵϕ0}|Δvλ|p(x)-2ΔvλΔϕdx.

Since the measure of the domain of integration {x:vλ+ϵϕ0} tends to zero as ϵ0+, it follows as ϵ0+ that

{x:vλ+ϵϕ0}|Δvλ|p(x)-2ΔvλΔϕdx0.

Dividing by ϵ and letting ϵ0+, we get

Ω|Δvλ|p(x)-2ΔvλΔϕdx-Ωg(x)vλ-γ(x)ϕdx-λΩf(x,vλ)ϕdx0.

Since the equality holds if we replace ϕ by -ϕ, which implies that vλ is a positive solution of problem (P+λ), this completes the proof of Theorem 3.3. ∎

6 An example

In this section, we give an example to illustrate our results.

Example 6.1.

Let Ω be a smooth bounded domain in N (N3), let p be a Lipschitz continuous function on Ω¯ with 1<p-p+<N2 and p*(x)=Np(x)N-2p(x), let s,s1,r and r1 be continuous functions on Ω¯ such that 1<max(r(x),r1(x))<p(x)<N2<min(s(x),s1(x)) for all xΩ, let γ:Ω¯(0,1) be a continuous function, let

gLp*(x)p*(x)+γ(x)-1(Ω),

and let h and h1 be two positive functions such that hLs(x)(Ω) and h1Ls1(x)(Ω). Put

f(x,t)={h(x)|t|β(x)-1,|t|1,h1(x)|t|α(x)-1,|t|>1,

with r(x)<β(x) and α(x)<r1(x) for all xΩ. Then conditions (I1), (I2) and (I3) are satisfied, so for any λ0 problem (P-λ) has a weak solution.

Moreover, if we suppose that f(x,t)=Ch(x)|t|r(x)-2t for all xΩ, then assumptions (f5) and (f6) hold, and consequently, for any λ0, problem (P+λ) has at least one nontrivial weak solution in W2,p(x)(Ω)W01,p(x)(Ω).

Acknowledgements

The authors would like to thank the anonymous referees for their suggestions and helpful comments which improved the presentation of the original manuscript.

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About the article

Received: 2016-12-01

Revised: 2017-11-11

Accepted: 2018-02-28

Published Online: 2018-03-13


Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 1171–1183, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2016-0260.

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© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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