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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


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An elliptic system with logarithmic nonlinearity

Claudianor Alves
  • Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, Av. Aprigio Veloso, 882, CEP 58429-900, Campina Grande /PB, Brazil
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/ Abdelkrim Moussaoui / Leandro Tavares
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  • Universidade Federal do Cariri, Av. Ten. Raimundo Rocha, CEP 63048-080, Juazeiro do Norte /CE, Brazil
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Published Online: 2017-11-27 | DOI: https://doi.org/10.1515/anona-2017-0200

Abstract

In the present paper, we study the existence of solutions for some classes of singular systems involving the Δp(x) and Δq(x) Laplacian operators. The approach is based on bifurcation theory and the sub-supersolution method for systems of quasilinear equations involving singular terms.

Keywords: Bifurcation; singular system; sub-supersolution

MSC 2010: 35J75; 35J48; 35J92

1 Introduction and statement of the main results

Let ΩN (N2) be a bounded domain with smooth boundary Ω. We are interested in the following quasilinear system:

{-Δp(x)u=-γlogv+θvα(x)in Ω,-Δq(x)v=-γlogu+θuβ(x)in Ω,u,v>0in Ω,u=v=0on Ω,(1.1)

which exhibits a singularity at zero through logarithm function. The variable exponents α(), β() are positive, the constants γ and θ are both greater than 0, and Δp(x) (resp. Δq(x)) stands for the p(x)-Laplacian (resp. q(x)-Laplacian) differential operator on W01,p(x)(Ω) (resp. W01,q(x)(Ω)) with p,qC1(Ω¯),

p(x)p(x),q(x)q(x),1<p-p+<N,1<q-q+<N,(1.2)

where p(x)=Np(x)N-p(x) and q(x)=Nq(x)N-q(x). In the sequel, we set

s-=infxΩs(x),s+=supxΩs(x)  for sC(Ω¯).

Throughout this paper, we denote by C1(Ω¯)×C1(Ω¯) the pair of functions (u,v)C1(Ω¯)×C1(Ω¯) such that there is a constant c>0, which depends on u and v, verifying

u(x),v(x)cd(x)in Ω,(1.3)

where d(x):=dist(x,Ω).

A weak solution of (1.1) is a pair (u,v)W01,p(x)(Ω)×W01,q(x)(Ω) with u, v being positive a.e. in Ω and satisfying

Ω|u|p(x)-2uϕdx=Ω(-γlogv+θvα(x))ϕ𝑑x,Ω|v|q(x)-2vψdx=Ω(-γlogu+θuβ(x))ψ𝑑x

for all (ϕ,ψ)W01,p(x)(Ω)×W01,q(x)(Ω).

The study of problems involving variable exponents growth conditions is widely justified with many physical examples, and these problems arise from a variety of nonlinear phenomena. They are used in electrorheological fluids as well as in image restorations. For more inquiries on modeling physical phenomena involving the p(x)-growth condition, we refer to [1, 2, 4, 6, 7, 14, 19, 21, 20, 23, 22].

Elliptic problems involving the logarithmic nonlinearity appear in some physical models like in the dynamic of thin films of viscous fluids; see for instance [12]. An interesting point regarding these problems comes from the fact that -logx is sign changing and behaving at the origin like the power function tα for α<0 with a slow growth. In addition, the logarithmic function is not invariant by scaling, which does not hold for the power function. These facts motivated the recent studies in [16, 8, 12], where de Queiroz et al. considered the scalar semilinear case of (1.1) (that is, p(x)=q(x)=2) with constant exponents and by essentially using the linearity of the principal part. We also mention [15], focusing on problems with constant exponents involving nonlinear operators.

The essential point in this work is that the singularity in system (1.1) comes through logarithmic nonlinearities involving variable exponents growth conditions. According to our knowledge, this happens for the first time when such problems are studied. Our main results provide the existence and regularity of (positive) solutions for problem (1.1). They are stated as follows.

Theorem 1.1.

Assume (1.2) holds.

  • (i)

    If

    0<α-α+<q--1,0<β-β+<p--1,(1.4)

    then problem ( 1.1 ) has a solution (u,v) for all θ,γ>0.

  • (ii)

    If

    α->q+-1,β->p+-1,(1.5)

    then problem ( 1.1 ) has a solution (u,v) for γ small enough and for all θ>0.

  • (iii)

    If

    α+>q--1,β+>p--1,(1.6)

    then problem ( 1.1 ) admits a solution (u,v) for γ and θ small enough.

Theorem 1.2.

Assume (1.2) and that

{γθe<α-α+<min{p--1,q(x)p(x)},γθe<β-β+<min{q--1,p(x)q(x)}(1.7)

holds for all xΩ, where e denotes the Euler number. Then problem (1.1) has a positive solution

(u,v)W01,p(x)(Ω)×W01,q(x)(Ω)

satisfying (1.3).

The proof of Theorem 1.1 is done in Section 4. Our approach relies on the sub-supersolutions techniques. However, this method in its system version (see [5, p. 269]) does not work for problem (1.1) due to its noncooperative character, which means that the right-hand sides of the equations in (1.1) are not necessarily increasing whenever u (resp. v) is fixed in the first (resp. second) equation in (1.1). Another reason this approach cannot be directly implemented is the presence of singularities in (1.1). To overcome this difficulties, we disturb problem (1.1) by introducing a parameter ε>0. This gives rise to a regularized system for (1.1) depending on ε>0, whose study is relevant for our initial problem. We construct a sub-supersolution pair for the regularized system independent on ε, and we show the existence of a positive family of solutions (uε,vε)C1,γ(Ω¯)×C1,γ(Ω¯), for certain γ(0,1), through a new result regarding sub-supersolutions for quasilinear competitive (noncooperative) systems involving variable exponents growth conditions (see Section 3). Then a (positive) solution of (1.1) is obtained by passing to the limit as ε0 essentially relying on the independence on ε of the upper and lower bounds of the approximate solutions (uε,vε) and on Arzelà–Ascoli’s Theorem. An important part of our result lies in obtaining the sub and supersolution, which cannot be constructed easily. Precisely, this is due to the fact that the p(x)-Laplacian operator is inhomogeneous and, in general, it has no first eigenvalue, that is, the infimum of the eigenvalues of the p(x)-Laplacian equals 0 (see [9]). At this point, the choice of suitable functions with an adjustment of adequate constants is crucial.

The proof of Theorem 1.2 is done in Section 5. It is chiefly based on a theorem by Rabinowitz (see [18]) which establishes, for each ε>0, the existence of positive solutions (uε,vε) for the regularized problem of (1.1) in W01,p(x)(Ω)×W01,q(x)(Ω). The solution of (1.1) under assumption (1.7) is obtained by passing to the limit as ε0. This is based on a priori estimates, the Hardy–Sobolev inequality and Lebesgue’s dominated convergence theorem.

A significant feature of our existence results concerns the regularity part. In Theorem 1.1, the regularity of the obtained solution for problem (1.1) is derived through the weak comparison principle and the regularity result in [3].

2 Preliminaries

Let pC(Ω¯) with p(x)>1 in Ω. Consider the Lebesgue space

Lp(x)(Ω):={u:Ω:u is measurable and Ω|u(x)|p(x)𝑑x<+},

which is a Banach space with the Luxemburg norm

uLp(x)(Ω):={λ>0:Ω|u(x)λ|p(x)𝑑x1}.

The Banach space W1,p(x)(Ω) is defined as

W1,p(x)(Ω):={uLp(x)(Ω):|u|Lp(x)(Ω)},

equipped with the norm

uW1,p(x)(Ω):=uLp(x)(Ω)+uLp(x)(Ω).

The space W01,p(x)(Ω) is defined as the closure of C0(Ω) in W1,p(x)(Ω) with respect to the norm. The space W01,p(x)(Ω) is a separable and reflexive Banach space when p->1. For a later use, we recall that the embedding

W01,p(x)(Ω)Lr(x)(Ω)

is compact with 1r(x)<p(x).

The following result gives important properties related to the logarithmic nonlinearity.

Lemma 2.1.

  • (i)

    For each α,θ>0 , there is a constant C that depends only on α and θ such that

    |log(x)|x-α+Cxθ

    for all x>0.

  • (ii)

    For each θ,ε>0 , there is a constant C that depends only on ε and θ such that

    |log(x+ε)|xθ+C

    for all x0.

  • (iii)

    Let γ, θ and δ be real numbers. If γ,θ>0 and δ>γθe , then the function f(x)=γxδ-θlogx, x>0 , attains a positive global minimum.

Proof.

With respect to the inequalities, we only prove (i) because (ii) can be justified similarly. A simple computation provides

limx0+|log(x)|x-α=0.

Thus, there is a small m>0 such that

|log(x)|x-αfor x(0,m).

On the other hand, the limit

limx+|log(x)|xθ=0

implies that there is M>0 such that

|log(x)|xθfor x(M,+).

Since the function |log(x)|/xθ, x>0, is continuous for all x>0, there is a constant which depends on α and θ such that |log(x)|Cxθ in [m,M]. Therefore, |log(x)|x-α+Cxθ for all x>0, where the constant C depends only on α and θ.

In order to show (iii), observe that f(x)=θδxδ-1-γx. Then f has a unique critical point at x0=(γθδ)1/δ. Thus, by solving the inequalities f(x)>0 and f(x)<0 for x>0, it follows that f is increasing on the interval [x0,+) and decreasing on (-,x0]. By noticing that

f(x0)=γδ(1-log(γθδ)),

the condition δ>γθe implies that f(x0)>0, which proves the result. ∎

3 Sub-supersolution theorem

Let us introduce the quasilinear system

{-Δp(x)u=H(x,u,v)in Ω,-Δq(x)v=G(x,u,v)in Ω,u=v=0on Ω,(3.1)

where H,G:Ω×+×+ are Carathéodory functions satisfying the following assumption:

  • (I)

    Given T,S>0, there is a constant C>0 such that

    |H(x,s,t)|,|G(x,s,t)|Cfor all (x,s,t)Ω×[0,T]×[0,S].

The following result is a key point in the proof of Theorem 1.1.

Theorem 3.1.

Assume that H and G satisfy (I), and let u¯W01,p(x)(Ω)L(Ω) and v¯W01,q(x)(Ω)L(Ω), with u¯,v¯0 in Ω and u¯,v¯W1,(Ω) such that

u¯u¯𝑎𝑛𝑑v¯v¯  in Ω.

Suppose that

Ω|u¯|p(x)-2uϕdxΩH(x,u¯,v¯)ϕ𝑑x,Ω|v¯|q(x)-2vψdxΩG(x,u¯,v¯)ψ𝑑x

and

Ω|u¯|p(x)-2uϕdxΩH(x,u¯,v¯)ϕ𝑑x,Ω|v¯|p(x)-2uϕdxΩH(x,u¯,v¯)ψ𝑑x

for all nonnegative functions (ϕ,ψ)W01,p(x)(Ω)×W01,q(x)(Ω). Then problem (3.1) has a (positive) solution

(u,v)(W01,p(x)(Ω)L(Ω))×(W01,q(x)(Ω)L(Ω))

satisfying

u¯(x)u(x)u¯(x)𝑎𝑛𝑑v¯(x)v(x)v¯(x)  for a.e. xΩ.

Proof.

The proof is chiefly based on pseudomonotone operator theory. Define the functions

H1(x,s,t)={H(x,u¯(x),v¯(x)),su¯(x),H(x,s,v¯(x)),u¯(x)su¯(x) and tv¯(x),H(x,s,t),u¯(x)su¯(x) and v¯(x)tv¯(x),H(x,s,v¯(x)),u¯(x)su¯(x) and tv¯(x),H(x,u¯(x),v¯(x)),su¯(x) and tv¯(x),

and

G1(x,s,t)={G(x,u¯(x),v¯(x)),tv¯(x),G(x,s,v¯(x)),v¯(x)tv¯(x) and su¯(x),G(x,s,t),v¯(x)tv¯(x) and u¯(x)su¯(x),G(x,s,v¯(x)),v¯(x)tv¯(x) and su¯(x),G(x,u¯(x),v¯(x)),tv¯(x).

In what follows, we fix l(0,1) with min{p-,q-}>1+l and set

γ1(x,s):=-((u¯(x)-s)+)l+((s-u¯(x))+)l,γ2(x,s):=-((v¯(x)-s)+)l+((s-v¯(x))+)l.

Using the above functions, we introduce the auxiliary problem

{-Δp(x)u=H2(x,u,v)in Ω,-Δq(x)v=G2(x,u,v)in Ω,u=v=0on Ω,(3.2)

where

H2(x,s,t):=H1(x,s,t)-γ1(x,s)(3.3)

and

G2(x,s,t):=G1(x,s,t)-γ2(x,s).

By the Minty–Browder theorem (see, e.g., [17]), problem (3.2) has a solution (u,v) in W01,p(x)(Ω)×W01,q(x)(Ω). Indeed, let B:EE be a function defined by

B(u,v),(ϕ,ψ):=Ω|u|p(x)-2uϕ+|v|q(x)-2vψdx-ΩH2(x,u,v)ϕ𝑑x-ΩG2(x,u,v)ϕ𝑑x,

where E is the Banach space W01,p(x)(Ω)×W01,q(x)(Ω) endowed with the norm

(u,v)=max{u1,p(x),v1,q(x)},(u,v)E.

Let us show that the function B satisfies the hypotheses of the Minty–Browder theorem.

(i) B is continuous.

Let (un,vn)E be a sequence that converges to (u,v) in E. We need to prove that B(un,vn)-B(u,v)E0. To this end, let (ϕ,ψ)E with (ϕ,ψ)E1. By the Hölder inequality, one has

|Ω|un|p(x)-2unϕ-|u|p(x)-2uϕdx|C|un|p(x)-2un-|u|p(x)-2uLp(x)p(x)-1(Ω).

Up to a subsequence, we can assume that un(x)u(x) a.e in Ω and that there exists a function U(Lp(x)(Ω))N such that |un(x)|U(x) a.e in Ω. Therefore, Lebesgue’s dominated convergence theorem yields

|un|p(x)-2un-|u|p(x)-2uLp(x)p(x)-1(Ω)0.

Note that

|Ω(H2(x,un,vn)-H2(x,u,v))ϕ𝑑x|Ω|H1(x,un,vn)-H1(x,u,v)||ϕ|𝑑x+Ω|γ1(x,un)-γ1(x,u)||ϕ|𝑑x.

Then the continuity and the boundedness of H, together with Lebesgue’s dominated convergence theorem and the Hölder inequality, gives

supϕ1Ω|H1(x,un,vn)-H1(x,u,v)||ϕ|𝑑x0.

On the other hand, we can assume that un(x)u(x) a.e in Ω and that there exists wLp(x)(Ω) such that |un(x)|w(x) a.e in Ω. Arguing as before, we get

γ1(x,u)-γ1(x,un)Lp(x)p(x)-1(Ω)0,

and so

supϕ1Ω(H2(x,un,vn)-H2(x,u,v))ϕ𝑑x0.

Hence, the previous reasoning provides

|vn|q(x)-2vn-|v|q(x)-2vLq(x)q(x)-1(Ω)0

and

supψ1Ω(G2(x,un,vn)-G2(x,u,v))ψ𝑑x0,

which justify the continuity of B.

(ii) B is bounded.

Let us show that if UE is a bounded set, then B(U)E is bounded. To this end, consider a bounded set U and (ϕ,ψ)E such that (ϕ,ψ)1. Then for (u,v)U the Hölder inequality gives

|Ω|u|p(x)-2uϕ+|u|q(x)-2vψdx|C(|u|p(x)-1Lp(x)p(x)-1(Ω)+|v|q(x)-1Lq(x)q(x)-1(Ω))C.

Since H1(x,u,v) is bounded, we derive that

Ω|H1(x,u,v)||ϕ|𝑑xCΩ|ϕ|𝑑xC.

On the other hand, since

Ω|γ1(x,u)|p(x)p(x)-1𝑑xCΩ(1+|u(x)|+|u¯(x)|+|u¯(x)|)p(x)p(x)-1𝑑x,

the Hölder inequality ensures

Ω|γ1(x,u)||ϕ|𝑑xC.

From the above arguments we obtain the boundedness of B.

(iii) B is coercive.

Next, we prove that

B(u,v),(u,v)(u,v)+as (u,v)+.

Note that

ΩH1(x,u,v)u𝑑x-Ω|H1(x,u,v)||u|𝑑x-CuLp(x)(Ω),(3.4)

where C is a positive constant. The triangular inequality and the fact that (a+b)θaθ+bθ for nonnegative numbers a and b with θ(0,1) give

Ω-γ1(x,u)udx={u¯u}(u¯-u)lu𝑑x-{uu¯}(u-u¯)lu𝑑x-{u¯u}(|u¯|+|u|)l|u|𝑑x-{uu¯}(u-u¯)lu𝑑x-{u¯u}(|u¯|l+|u|l)|u|𝑑x-{uu¯}{u>0}(u-u¯)lu𝑑x-{uu¯}{u<0}(u-u¯)lu𝑑x-Ω(|u¯|l+|u|l)|u|𝑑x-Ω(|u¯|l+|u|l)|u|𝑑x.

Gathering the last inequality with the embeddings

W1,p(x)(Ω)Lp(x)(Ω)andLp(x)(Ω)L1+l(Ω),

we derive

Ωγ1(x,u)u𝑑x-CuLp(x)(Ω)-Ω|u|1+l𝑑x-CuLp(x)(Ω)-CuLp(x)(Ω)1+l.(3.5)

From (3.3)–(3.5) we have

-ΩH2(x,u,v)u𝑑x-CuLp(x)(Ω)-CuLp(x)(Ω)1+l-C(u,v)-C(u,v)1+l,

where C is a positive constant. In the same manner, we can see that

-ΩG2(x,u,v)u𝑑x-CvLq(x)(Ω)-CvLq(x)(Ω)1+l-C(u,v)-C(u,v)1+l.

  • If uLp(x)(Ω)1 and vLq(x)(Ω)<1, then

    Ω|u|p(x)𝑑x+Ω|v|q(x)𝑑xuLp(x)(Ω)p-+vLq(x)(Ω)q+

  • If uLp(x)(Ω)1 and vLq(x)(Ω)1, then

    Ω|u|p(x)𝑑x+Ω|v|q(x)𝑑xuLp(x)(Ω)p-+uLq(x)(Ω)q-.

Consider in E a sequence {(un,vn)}n such that (un,vn)+. Thus,

unLp(x)(Ω)+orvnLq(x)(Ω)+.

Suppose that the first possibility happens and that unLp(x)(Ω)1 for all n. Then we consider two cases:

  • unLp(x)(Ω)1 and vnLq(x)(Ω)<1 for n. In this case, we have

    B(un,vn),(un,vn)(un,vn)EunLp(x)(Ω)p-+vnLq(x)(Ω)q+unLp(x)(Ω)-CvnLq(x)(Ω)1+lunLp(x)(Ω)-C-CunLp(x)(Ω)lunLp(x)(Ω)p--1-CunLp(x)(Ω)-C-unLp(x)(Ω)l.

  • unLp(x)(Ω)1 and vnLq(x)(Ω)1 for n. In this second case, we have

    B(un,vn),(un,vn)(un,vn)E(max{unLp(x)(Ω),vnLq(x)(Ω)})min{p-,q-}max{unLp(x)(Ω),vnLq(x)(Ω)}-C-C(max{unLp(x)(Ω),vnLq(x)(Ω)})l.

Consequently, in both cases studied above, one has

B(un,vn),(un,vn)(un,vn)Eas n+.

The other situations regarding unLp(x)(Ω) and vnLq(x)(Ω) can be handled in much the same way.

(iv) B is pseudomonotone.

We recall that B is a pseudomonotone operator if (un,vn)(u,v) in E and

lim supn+B(un,vn),(un,vn)-(u,v)0.(3.6)

Then

lim infn+B(un,vn),(un,vn)-(ϕ,ψ)B(u,v),(u,v)-(ϕ,ψ)

for all (ϕ,ψ)E.

If (un,vn)(u,v), then unu and vnv in W1,p(x)(Ω) and W1,q(x)(Ω), respectively. Since H1 and G1 are bounded, we must have

ΩH1(x,un,vn)(un-u)𝑑x0

and

ΩG1(x,un,vn)(un-u)𝑑x0.

Note that

B(un,vn),(un,vn)-(u,v)=Ω|un|p(x)-2un,un-u𝑑x+ΩH2(x,un,vn)(un-u)𝑑x+Ω|vn|q(x)-2vn,vn-v𝑑x+ΩG2(x,un,vn)(vn-v)𝑑x.

The previous arguments can be repeated to show that

ΩH2(x,un,vn)(un-u)𝑑x=Ω[H1(x,un,vn)(un-u)-γ1(x,un)(un-u)]𝑑x,limn+ΩH2(x,un,vn)(un-u)𝑑x=0,limn+ΩG2(x,un,vn)(vn-v)𝑑x=0.

Gathering the above limits together with (3.6), we have

lim¯Ω|un|p(x)-2un,(un-u)𝑑x+Ω|vn|q(x)-2vn,(vn-v)𝑑x0.(3.7)

From the weak convergence we get

Ω|u|p(x)-2u,(un-u)𝑑x=on(1)

and

Ω|v|p(x)-2v,(vn-v)𝑑x=on(1).

Therefore,

Ω|un|p(x)-2un,(un-u)𝑑x=Ω|un|p(x)-2un-|u|p(x)-2u,(un-u)𝑑x+on(1)(3.8)

and

Ω|vn|p(x)-2vn,(vn-v)𝑑x=Ω|vn|p(x)-2vn-|v|p(x)-2v,(vn-v)𝑑x+on(1).(3.9)

By using (3.8) and (3.9) in (3.7), the (S+) property of the operators -Δp(x) and -Δq(x) guarantees that unu in W01,p(x)(Ω) and vnv in W01,q(x)(Ω). Thus, by the continuity of B, it turns out that

limn+B(un,vn),(un,vn)-(ϕ,ψ)=B(u,v),(u,v)-(ϕ,ψ)

for all (ϕ,ψ)E.

Finally, from properties (i)–(iv) we are in a position to apply [17, Theorem 3.3.6] which ensures that B is surjective. Thereby, there exists (u,v)E such that

B(u,v),(ϕ,ψ)=0for all (ϕ,ψ)E,

and, in particular, (u,v) is a solution of (3.2).

It remains to prove that

u¯uu¯andv¯vv¯  in Ω.(3.10)

We only prove the first inequalities in (3.10) because the second ones can be justified similarly. Set (ϕ,ψ):=((u-u¯)+,0). From the definition of H2 we obtain

Ω|u|p(x)-2u(u-u¯)+={uu¯}H1(x,u,v)(u-u¯)+dx-{uu¯}(-((u¯-u)+)l+((u-u¯)+)l)(u-u¯)+dx={uu¯}H(x,u¯,v¯)(u-u¯)+𝑑x-Ω((u-u¯)+)l+1𝑑xΩ|u¯|p(x)-2u¯(u-u¯)+dx-Ω((u-u¯)+)l+1dx.

Therefore,

Ω|u|p(x)-2u,|u¯|p(x)-2u¯,(u-u¯)+dx-Ω((u-u¯)+)l+1dx0,

wich implies that uu¯ in Ω. Using a quite similar argument for (ϕ,ψ):=((u¯-u)+,0), we get u¯u in Ω. This completes the proof. ∎

4 Proof of Theorem 1.1

For every ε>0, let us introduce the auxiliary problem

{-Δp(x)u=-γlog(|v|+ε)+θ|v|α(x)in Ω,-Δq(x)v=-γlog(|u|+ε)+θ|u|β(x)in Ω,u=v=0on Ω.(4.1)

Our goal is to show through Theorem 3.1 that (4.1) has a positive solution (uε,vε). Then, by passing to the limit as ε0+, we get a solution for the original problem (1.1).

Let Ω~ be a bounded domain in N with smooth boundary Ω~ such that Ω¯Ω~, and set d~(x)=dist(x,Ω~). In [24, Lemma 3.1], Yin and Zang have proved that, for δ>0 small enough and for constants η>0, the function

wC1(Ω~¯)C0(Ω~)

defined by

w(x)={ξd~(x)if d~(x)<δ,ξδ+δd~(x)ξ(2δ-tδ)2p--1if δd~(x)<2δ,ξδ+δ2δξ(2δ-tδ)2p--1if 2δd~(x),

is a subsolution of the problem

{-Δp(x)u=ηin Ω~,u=0on Ω~,

where δ>0 is a number that does not depend on η, and

ξ=c0η1p+-1+τ,

with a fixed number τ(0,1) and c0>0 is a number depending only on δ, τ, Ω~ and p. Note that

w(x)=c0η1p+-1+τd~(x)for d~(x)<δ,c0η1p+-1+τδw(x)for d~(x)δ.

Given λ>1, let u¯ and v¯ in C1(Ω~¯) be the unique solutions of the problems

{-Δp(x)u¯=λσin Ω~,u¯=0on Ω~,and{-Δq(x)v¯=λσin Ω~,v¯=0on Ω~,(4.2)

where σ is a real constant.

If σ>0, considering the corresponding function w for η=λσ and applying the weak maximum principle, we get

{C0λσp+-1+τ1min{δ,d~(x)}u¯(x)C1λσp--1,C0λσq+-1+τ2min{δ,d~(x)}v¯(x)C1λσq--1,in Ω~,(4.3)

where C0,C0,C1,C1>0 and τ1,τ2(0,1) are constants that do not depend on λ. If -1<σ<0, from [11, Lemma 2.1] and for λ large one has

u¯(x)k2λσp--1c2λσp+-1andv¯(x)k2λσq--1c2λσq+-1  in Ω~,(4.4)

where k2, k2, c2 and c2 are positive constants independent of λ. Moreover, by the strong maximum principle there is a constant c0>0 (that can depend on λ) such that

c0d~(x)min{u¯(x),v¯(x)}(4.5)

Now, let u¯ and v¯ in C1(Ω¯) be the unique solutions of the homogeneous Dirichlet problems

{-Δp(x)u¯=λ-1in Ω,u¯=0on Ω,and{-Δq(x)v¯=λ-1in Ω,v¯=0on Ω.(4.6)

By [10, Lemma 2.1] and [11], there exist positive constants k0, K1 and K2 independent of λ such that

u¯(x)K1λ-1p--1andv¯(x)K2λ-1q--1  in Ω(4.7)

and

k0d(x)min{u¯(x),v¯(x)}in Ω.(4.8)

By the weak maximum principle, we have u¯u¯ and v¯v¯ in Ω¯ for λ>1 sufficiently large.

We state the following existence result for the regularized problem (4.1).

Theorem 4.1.

Under the assumptions of Theorem 1.1, there exists ε0>0 such that system (4.1) has a positive solution

(uε,vε)(W01,p(x)(Ω)L(Ω))×(W01,q(x)(Ω)L(Ω))

for all ε(0,ε0). Moreover,

u¯(x)uε(x)u¯(x)𝑎𝑛𝑑v¯(x)vε(x)v¯(x)  for a.e. xΩ.(4.9)

Proof.

First, let us show that (u¯,v¯) is a subsolution for problem (4.1) for all ε(0,ε0). To this end, pick ε012. Then from (4.6) and (4.7), for all ε(0,ε0), one has

-Δp(x)u¯=λ-1-γlog(K2λ-1q--1+ε0)-γlog(v¯(x)+ε)-γlog(v¯(x)+ε)+θv¯(x)α(x)in Ω

and

-Δq(x)v¯=λ-1-γlog(K1λ-1p--1+ε0)-γlog(u¯(x)+ε)-γlog(u¯(x)+ε)+θu¯(x)β(x)in Ω

for all γ,θ>0, provided that λ>1 is sufficiently large.

Next, we will show that (u¯,v¯) is a supersolution for problem (4.1) for all ε(0,ε0). Set δ:=dist(Ω~,Ω) and fix ε0(0,1). By Lemma 2.1, there are constants σ1,σ2(0,1) and Cσ1,α+,Cσ2,β+>0 such that, for all ε(0,ε0), one has

-γlog(v¯+ε)+θv¯α(x)γ(1(v¯+ε)σ1+Cσ,1α+(v¯+ε)α+)+θv¯α(x)(4.10)

and

-γlog(u¯+ε)+θu¯β(x)γ(1(u¯+ε)σ2+Cσ2,β+(u¯+ε)β+)+θu¯β(x).(4.11)

If (1.4) holds, it follows from (4.3), (4.10), (4.11) and for σ>0 in (4.2) that

-γlog(v¯+ε)+θv¯α(x)γ(1v¯σ1+Cσ1,α+(v¯+ε0)α+)+θ(v¯+1)α+γv¯σ1+2α+-1(γCσ1,α++θ)(v¯α++1)γ(v¯)σ1+2α+-1(γCσ1,α++θ)C1(λσα+q--1+1)γ(C0λσq+-1+τ2min{δ,δ})σ1+2α+-1(γCσ1,α++θ)(C1λσα+q--1+1)λσin Ω(4.12)

and

-γlog(u¯+ε)+θu¯β(x)γ(1u¯σ2+Cσ2,β+(u¯+ε)β+)+θ(u¯+1)β+γu¯σ2+2β+-1(γCσ2,β++θ)(u¯β++1)γ(C0λσp+-1+τ1min{δ,δ})σ2+2β+(γCσ2,α++θ)(C1λσβ+p--1+1)λσin Ω(4.13)

for all γ,θ>0, provided that λ is large enough.

If (1.5) is satisfied, combining Lemma 2.1 with (4.4) and (4.5), by (4.10), (4.11) and for σ(-1,0) in (4.2), we get

-γlog(v¯+ε)+θv¯α(x)γ(1(v¯+ε)σ1+Cσ,1α-(v¯+ε)α-)+θv¯α(x)γ(1v¯σ1+Cσ1,α-2α--1v¯α-+Cσ1,α-ε0α-2α--1)+θv¯α(x)γ(1(c0δ)σ1+Cσ1,α-2α--1λσα-q+-1+Cσ1,α-2α--1)+θc2λσα(x)q+-1γ(1(c0δ)σ1+Cσ1,α-2α--1λσα-q+-1+Cσ1,α-2α--1)+θc2λσα-q+-1λσin Ω(4.14)

and

-γlog(u¯+ε)+u¯β(x)γ(c0δ)σ1+γCσ1,β-2β--1λσα-q+-1+γCσ1,β-2β--1+c2λσβ-p+-1λσin Ω(4.15)

for γ>0 small enough, for all θ>0 and all ε(0,ε0), provided that λ is sufficiently large.

Finally, if (1.6) holds, using (4.3), (4.5), (4.10) and (4.11), for σ(-1,0) in (4.2), we obtain

-γlog(v¯+ε)+θv¯α(x)γ(1(v¯+ε)σ1+Cσ1,α+(v¯+1)α+)+θv¯α(x)γ(1v¯σ1+Cσ1,α+(v¯α++1))+θ(v¯+1)α+γ(1(c0δ)σ1+Cσ1,α+(v¯α++1))+2α+θ(v¯α++1)γ(1(c0δ)σ1+Cσ1,α+(λσα+q--1+1))+2α+θ(λσα+q--1+1)λσin Ω¯,(4.16)

and similarly

-γlog(u¯+ε)+θu¯β(x)γ(1u¯σ2+Cσ2,β+(u¯+1)β+)+θu¯β(x)λσin Ω¯(4.17)

for all γ,θ>0 small and all ε(0,ε0), provided that λ>0 is large enough.

Consequently, it turns out from (4.12), (4.13), (4.14), (4.15), (4.16) and (4.17) that

Ω|u¯|p(x)-2u¯ϕdxΩ(-γlog(v¯+ε)+θv¯α(x))ϕ𝑑x

and

Ω|v¯|q(x)-2v¯ψdxΩ(-γlog(u¯+ε)+θu¯α(x))ψ𝑑x

for all (ϕ,ψ)W01,p(x)(Ω)×W01,q(x)(Ω) with ϕ,ψ0. This shows that (u¯,v¯) is a supersolution for (4.1) for all ε(0,ε0).

Then, owing to Theorem 3.1, we conclude that the perturbed problem (4.1) has a solution

(uε,vε)W01,p(x)(Ω)×W01,q(x)(Ω)

within [u¯,u¯]×[v¯,v¯] for all ε(0,ε0). Moreover, according to Lemma 2.1 combined with (4.8) and (4.9), we have that for σ1,σ2(0,1N) there are constants Cσ1,Cσ2>0 such that

-γlog(vε+ε)+θvεα(x)γ(vε-σ1+Cσ1vεσ1)+θvεα(x)=vε-σ1(γ+γCσ1vε2σ1+θvεα(x)+σ1)v¯-σ1(γ+γCσ1v¯2σ1+θv¯α(x)+σ1)(k0d(x))-σ1(γ+γCσ1v¯2σ1+θv¯α(x)+σ1)A1d(x)-σ1in Ω

and

-γloguε+uεβ(x)γ(uε-σ2+Cσ2uεσ2)+θuεβ(x)=u¯-σ2(γ+γCσ2u¯2σ2+θu¯β(x)+σ2)(k0d(x))-σ2(γ+γCσ2u¯2σ2+θu¯β(x)+σ2)A2d(x)-σ2in Ω

for some positive constants A1 and A2. Then, thanks to [3, Lemma 2], we deduce that

(uε,vε)C1,ν(Ω¯)×C1,ν(Ω¯)

for certain ν(0,1). ∎

Proof of Theorem 1.1.

Set ε:=1n for n1/ε0. By Theorem 4.1, we know that there exists a positive solution (un,vn):=(u1/n,v1/n) bounded in C1,ν(Ω¯)×C1,ν(Ω¯), for certain ν(0,1), for the problem

{-Δp(x)un=-γlog(|vn|+1n)+θ|vn|α(x)in Ω,-Δq(x)vn=-γlog(|un|+1n)+θ|un|β(x)in Ω,un=vn=0on Ω.

Moreover, the property formulated in (4.9) holds true. Employing Arzelà–Ascoli’s theorem, we may pass to the limit in C1(Ω¯)×C1(Ω¯) and the limit functions (u,v)C1(Ω¯)×C1(Ω¯) satisfy (1.1) with (u,v)[u¯,u¯]×[v¯,v¯]. This completes the proof. ∎

5 Proof of Theorem 1.2

This section is devoted to the proof of Theorem 1.2. For ε>0, let us consider the regularized problem

{-Δp(x)u=-γlog(|v|+ε)+θ(|v|+ε)α(x)in Ω,-Δq(x)v=-γlog(|u|+ε)+θ(|u|+ε)β(x)in Ω,u=v=0on Ω.(5.1)

Our demonstration strategy will be to show, by applying the well-known result due to Rabinowitz [18], that for each λ>0 system (5.1) possesses a positive solution (uε,vε) in W01,p(x)(Ω)×W01,q(x)(Ω), and then derive a solution of (1.1) by taking the limit ε0.

5.1 Existence result for the regularized system

Fix ε>0, and for each pair (f,g)Lp(x)(Ω)×Lq(x)(Ω) let us consider the auxiliary problem

{-Δp(x)u=λ(-γlog(|g|+ε)+θ(|g|+ε)α(x))in Ω,-Δq(x)v=λ(-γlog(|f|+ε)+θ(|f|+ε)β(x))in Ω,u=v=0on Ω.(5.2)

Observe the following facts:

  • -log(|g|+ε)Lp(x)(Ω): Indeed, consider θ>0 such that 0<θ(p)+q(x) for all xΩ¯. By Lemma 2.1, one has

    |log(|g(x)|+ε)|p(x)(|g(x)|θp(x)+C)((1+|g(x)|)θp(x)+C).

    From (1.2) the claim follows.

  • (|g(x)|+ε)α(x)Lp(x): By (1.2), notice that

    (|g(x)|+ε)α(x)p(x)(|g(x)|+1)α(x)p(x)(|g(x)|+1)q(x).

    Since W1,q(x)(Ω)Lq(x)(Ω), the claim is proved.

In the same manner, we have |log(|f|+ε)|Lq(Ω) and (|f|+ε)β(x)Lq(x) for all fLp(x)(Ω). Then, on account of the above remarks, the unique solvability of (u,v)W01,p(x)(Ω)×W01,q(x)(Ω) in (5.2) is readily derived from the Minty–Browder theorem. Therefore, the solution operator

𝒯:+×Lp(x)(Ω)×Lq(x)(Ω)W01,p(x)(Ω)×W01,q(x)(Ω)

is well defined.

Lemma 5.1.

The operator T:R+×Lp(x)(Ω)×Lq(Ω)Lp(x)(Ω)×Lq(x)(Ω) is continuous and compact.

Proof.

Consider a sequence (λn,fn,gn)(λ,f,g) in +×Lp(x)(Ω)×Lq(x)(Ω), and (un,vn):=𝒯(λn,fn,gn). By using un as a test function, one gets

Ω|un|𝑑x=λn(Ω-γunlog(|gn|+ε)+θun(|gn|+ε)α(x))dxCunLp(x)(Ω)log(|gn|+ε)Lp(x)(Ω)+CunLp(x)(Ω)(|gn|+ε)α(x)Lp(x)(Ω).(5.3)

Since {gn} is bounded in Lq(x)(Ω), by Lemma 2.1, {un} is bounded in W1,p(x)(Ω). Let (u,v):=𝒯(λ,f,g). Using un-u as a test function, we have

Ω|un|p(x)-2un-|u|p(x)-2u,(un-u)𝑑x=γΩ(λlog(|g|+ε)-λlog(|gn|+ε))(un-u)𝑑x   +θΩ(λ(|g|+ε)α(x)-λn(|gn|+ε)α(x))(un-u)𝑑x.(5.4)

Note that

|γΩ(λlog(|g|+ε)-λnlog(|gn|+ε))(un-u)𝑑x|(5.5)Cun-uLp(x)(Ω)(λlog(|g|+ε)-λnlog(|gn|+ε))Lp(x)(Ω),(5.6)

where the constant C does not depend on n.

In the sequel, up to a subsequence, we can assume that gn(x)g(x) a.e in Ω and |gn(x)|h a.e in Ω for some hLq(x)(Ω). Then, by Lemma 2.1 and the Lebesgue theorem, we have

γΩ(λlog(|g|+ε)-λnlog(|gn|+ε))(un-u)𝑑x0.(5.7)

A similar reasoning leads to

θΩ(λ(|g|+ε)α(x)-λn(|gn|+ε)α(x))(un-u)𝑑x0.(5.8)

Since {un} is bounded in W01,p(x)(Ω), from (5.4) we deduce that unu in W01,p(x)(Ω). This proves that 𝒯 is continuous.

In order to show that 𝒯 is compact, it suffices to prove that 𝒯(U)¯ is compact for all UE bounded. At this point, a quite similar argument as above produces the desired conclusion. This completes the proof. ∎

Theorem 5.2.

Under assumptions (1.2) and (1.7), problem (5.1) admits a solution (uε,vε) for all ε>0.

Proof.

From Lemma 5.1 and by invoking [18], there is an unbounded continuum 𝒞 of solutions of the equation (u,v)=𝒯(λ,u,v), that is, (λ,u,v)𝒞 is a solution of (5.1).

On the other hand, by Lemma 2.1, the function f(x)=θxδ-γlogx, for x>0, attains a strictly positive minimum if δ>γθe. Since α-,α+>γθe, we obtain the following assertions:

  • If |u|+ε1, then

    -γlog(|u|+ε)+θ(|u|+ε)α(x)-γlog(|u|+ε)+θ(|u|+ε)α-m->0,

    where m-=min{-γlogx+θxα-:x>0}.

  • If |u|+ε<1, then

    -log(|u|+ε)+(|u|+ε)α(x)-log(|u|+ε)+(|u|+ε)α+m+>0,

    where m+=min{-γlogx+θxα+:x>0}.

Therefore, -Δp(x)um1>0, where m1=min{m-,m+}, and, with a quite similar reasoning, we get -Δq(x)vm2>0 for some m2>0. Thus, by the maximum principle, 𝒞{(0,0,0)} must be constituted by strictly positive functions.

Next, we show that the component 𝒞 is unbounded with respect to λ0. By contradiction, suppose that there is λ>0 such that (λ,u,v)𝒞 implies that λλ. Fix 0<γ¯(q/p)-. Using u as a test function, we get

Ω|u|p(x)𝑑x=λ(Ω-γulog(|v|+ε)dx+Ωθ(|v|+ε)α(x)u𝑑x)(5.9)

and

Ω|-γulog(|v|+ε)|𝑑xΩγ|u|((|v|+ε)-γ¯+Cγ,γ¯(|v|+ε)γ¯)𝑑xCuLp(x)(Ω)+CΩ|u||v|γ¯𝑑x,(5.10)

where C depends on λ, ε, γ and γ¯. Note that

Ωθ|u|(|v|+ε)α(x)𝑑xΩθ|u|(|v|+1)α(x)𝑑xCuLp(x)(Ω)+CΩ|u||v|α(x)𝑑x,(5.11)

where C depends on θ and ε. Now we will estimate the integral Ω|u||v|α(x)𝑑x. We have |v(x)|α(x)Lp(x)(Ω). In order to prove this, note that

|v(x)|α(x)p(x)(1+|v(x)|)α(x)p(x)(1+|v(x)|)q(x).

The last function belongs to L1(Ω) because W1,q(x)(Ω)Lq(x)(Ω)Lq(x)(Ω). Thus, by the Hölder inequality we obtain

Ω|v|α(x)|u|𝑑xC|v|α(x)Lp(x)(Ω)uW1,p(x)(Ω).

By the Hölder inequality and considering all the possibilities for the norms

|v|α(x)Lp(Ω)and|v|α(x)Lq(x)α(x)p(x)(Ω),

we get

|v|α(x)Lp(x)(Ω)C|v|α(x)p(x)Lq(x)α(x)p(x)(Ω)1(p)++1+C|v|α(x)p(x)Lq(x)α(x)p(x)(Ω)1(p)-C((Ω|v|q(x)𝑑x)1(p)+(α(x)p(x)q(x))++1+(Ω|v|q(x)𝑑x)1(p)+(α(x)p(x)q(x))-)+C((Ω|v|q(x)𝑑x)1(p)-(α(x)p(x)q(x))++1+(Ω|v|q(x)𝑑x)1(p)-(α(x)p(x)q(x))-).(5.12)

Using the embedding W1,q(x)(Ω)Lq(x)(Ω), considering all the possibilities for the norms vLq(x)(Ω) and uLp(x)(Ω) and estimates (5.9), (5.10), (5.11) and (5.12), and repeating the arguments for the integral Ω|u||v|γ𝑑x, we obtain

uW1,p(x)(Ω)CuW1,p(x)(Ω)1p-(1+vW1,q(x)(Ω))α+p-+CuW1,p(x)(Ω)1p+(1+vW1,q(x)(Ω))α+p-+CuW1,p(x)(Ω)1p-+CuW1,p(x)(Ω)1p+.

Thus,

uW1,p(x)(Ω)p-CuW1,p(x)(Ω)(1+vnW1,p(x)(Ω))α++C(1+uW1,p(x)(Ω)).(5.13)

A similar reasoning leads to

vW1,p(x)(Ω)q-CvW1,q(x)(Ω)(1+uW1,p(x)(Ω))β++C(1+vW1,q(x)(Ω)).(5.14)

Since α++1<p- and β++1<q-, it follows that the component 𝒞 is bounded, which is absurd. Consequently, 𝒞 crosses the set {1}×Lp(x)(Ω)×Lq(x)(Ω), and this implies that there is a solution (uε,vε) of (5.1). The proof is completed. ∎

5.2 Passage to the limit

Set ε=1n in (5.1) with any integer n1. By applying Theorem 5.2, we know that there exist u1/n:=un and v1/n:=vn that solve the problem (5.1) with ε=1n.

Claim.

The sequences {un} and {vn} are bounded in W01,p(x)(Ω) and W01,q(x)(Ω), respectively, and the weak limits (that exist up to a subsequence) are strictly positive in Ω.

First of all, we know that -Δp(x)unm1>0, where m1=min{m-,m+}. If w1 denotes the unique positive solution of

{-Δp(x)w1=m1in Ω,w1=0on Ω,

the maximum principle gives

unw1>0in Ω.

By the strong maximum principle (see [11, Theorem 1.2]), we have w1η>0, where η is the inward normal vector of Ω. Let ϕq- be an eigenfunction associated to the first eigenvalue of the operator (-Δq-,W01,q-(Ω)). Note that

P1ϕq-(x)w1(x),

where P1 is a positive constant that does not depend on xΩ.

Denote by ϕp- an eigenfunction associated to the first eigenvalue of the operator (-Δp-,W01,p-(Ω)). Reasoning as above, we also have vnw2>0 and

L1ϕp-(x)w2(x),(5.15)

where L1 is a positive constant that does not depend on xΩ, with w2 being the unique positive solution of

{-Δq(x)w2=m2in Ω,w2=0on Ω.

Let δ(0,1). By using un as a test function in its corresponding system of equations and arguing as in the set of inequalities (5.10) and (5.11), we get

Ω|un|p(x)𝑑xC(Ω|un||vn|δ𝑑x+Ω|un||vn|δ𝑑x+unLp(x)(Ω))+CΩ|un||vn|α(x)𝑑x,

where C is a constant that depends on γ and θ. By the Hardy–Sobolev inequality (see [13]), together with the embedding W01,p(x)(Ω)W01,p-(Ω) and the relation (5.15), it follows that

Ω|un||vn|δ𝑑xΩ|un|w2δ𝑑xΩ|un|Cϕp-δ𝑑xCunLp(x)(Ω),(5.16)

where the constant C does not depend on n.

By (5.16) and using the reasoning that leads to (5.13) and (5.14), we obtain that (un,vn) is bounded in E. Passing to a subsequence, we have

  • unu in W1,p(x)(Ω),

  • unu in Lp(x)(Ω),

  • unu a.e in Ω,

  • vnv in W1,p(x)(Ω),

  • vnv in Lp(x)(Ω),

  • vnv a.e in Ω

for some pair (u,v)E. From the previous pointwise convergence and the relations between w1, un and w2, vn, we conclude that u>0 and v>0, which proves the claim.

Taking un as a test function and repeating the arguments of relations (5.3)–(5.8), we get that unu in W01,p(x)(Ω). Notice that the same argument provides that vnv in W01,q(x)(Ω).

From the previous strong convergence of un and vn, combined with Lebesgue’s dominated convergence theorem, we obtain

Ω|u|p(x)-2uϕdx=Ω-γϕlogvdx+Ωθϕv𝑑x

and

Ω|v|p(x)-2uψdx=Ω-γψlogudx+Ωθψu𝑑x

for all (ϕ,ψ)W01,p(x)(Ω)×W01,q(x)(Ω), and the existence of a solution is proved.

Acknowledgements

The authors thank the anonymous referees and the editor, Prof. Vicentiu Rădulescu, for their valuable comments which helped to improve this work. The work was started while the second and the third authors were visiting the Federal University of Campina Grande. They thank Prof. Claudianor Alves and the other members of the department for hospitality.

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About the article

Received: 2017-09-11

Revised: 2017-10-11

Accepted: 2017-10-13

Published Online: 2017-11-27


Funding Source: Conselho Nacional de Desenvolvimento Científico e Tecnológico

Award identifier / Grant number: 304036/2013-7

Award identifier / Grant number: 402792/2015-7

The first author was partially supported by Conselho Nacional de Desenvolvimento Científico (304036/2013-7) and Instituto Nacional de Ciência de Tecnologia. The second author was supported by Conselho Nacional de Desenvolvimento Científico (402792/2015-7).


Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 928–945, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0200.

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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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