# Multiple Solutions for a Class of Differential Inclusion System Involving the(p(x),q(x))-Laplacian

Authors: Bin Ge; Ji-Hong Shen
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We consider a differential inclusion system involving the(p(x),q(x))-Laplacian with Dirichlet boundary condition on a bounded domain and obtain two nontrivial solutions under appropriate hypotheses. Our approach is variational and it is based on the nonsmooth critical point theory for locally Lipschitz functions.


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## Body

## 1. Introduction

In recent years, the study of differential equations and variational problems withp(x)-growth conditions has been a new and interesting topic, which arises from nonlinear electrorheological fluids (see [1]) and elastic mechanics (see [2]). The study on variable exponent problems attracts more and more interest in recent years, and many results have been obtained on this kind of problems, for example [3–11].Elliptic systems with standard growth conditions have been the subject of a sizeable literature. We refer to the excellent survey article of de Figueiredo [12].In [11], the author obtained the existence and multiplicity of solutions for the following problem:(P1)-div⁡(|∇u|p(x)-2∇u)=Fu(x,u,v),inΩ,-div⁡(|∇v|q(x)-2∇v)=Fv(x,u,v),inΩ,u=v=0,on⁡∂Ω,
where Ω⊂ℝN is a bounded domain with a smooth boundary ∂Ω, N≥2, (p,q)∈C(Ω¯)2, p(x)>1, q(x)>1, for every x∈Ω. The function F is assumed to be continuous in x∈Ω and of class C1 in u,v∈ℝ. More precisely, the author was able to prove that, under suitable conditions, the system might have at least one solution or have infinite number of solutions.Since many free boundary problems and obstacle problems may be reduced to partial differential equations with discontinuous nonlinearities, now a question arises: whether there exist solutions for system (P1) in the case where there is no continuously differentiable hypothesis required on the potential function F with respect to t(s). See, for example, F(x,t,s) is locally Lipschitz with respect to t(s). That is the main problem which we want to solve in the present paper.To this end, we mainly discuss the existence and multiplicity of solutions for the following nonlinear differential inclusion system involving the(p(x),q(x))-Laplacian:(P)-div⁡(|∇u|p(x)-2∇u)∈λ∂uF(x,u,v),inΩ,-div⁡(|∇v|q(x)-2∇v)∈λ∂vF(x,u,v),inΩ,u=v=0,on⁡∂Ω,
where Ω⊂ℝN is a bounded domain with C1-boundary ∂Ω, λ>0 is the parameter, p,q∈C(Ω¯), 1<p-≤p+<+∞, 1<q-≤q+<+∞, F:Ω×ℝ×ℝ→ℝ is a function such that F(·,t,s) is measurable in Ω for all (t,s)∈ℝ×ℝ, and F(x,t,s) is locally Lipschitz with respect to t(s) (in general it can be nonsmooth), ∂tF(x,t,s)(∂sF(x,t,s)) is the subdifferential with respect to the t(s)-variable in the sense of Clarke [13].We emphasize that the operator-div(|∇u|p(x)-2∇u) is said to be p(x)-Laplacian, which becomes p-Laplacian when p(x)≡p (a constant). The p(x)-Laplacian possesses more complicated nonlinearities than the p-Laplacian, for example, it is inhomogeneous and, in general, it does not have the first eigenvalue. In other words, the infimum of the eigenvalues of p(x)-Laplacian equals 0 (see [14]).Specially, ifF(x,·,v),F(x,u,·)∈C1(ℝ) for a.a. x∈Ω, and p(x)=p, q(x)=q, then the problem (P) becomes the following problem:(P2)-div⁡(|∇u|p-2∇u)=λFu(x,u,v),inΩ,-div⁡(|∇v|q-2∇v)=λFv(x,u,v),inΩ,u=v=0,on⁡∂Ω.
There have been a large number of papers that study the existence of the solutions to (P2). For instance, when p>N,q>N, Li and Tang [15] ensured the existence of three solutions to this problem. In [16], Kristály studied the multiplicity of solutions of the quasilinear elliptic systems (P1), where Ω is a strip-like domain and λ>0 is the parameter. Under some growth conditions on F, the author guaranteed the existence of an open interval Λ⊂[0,+∞), such that for each λ∈Λ problem (P2) has at least two distinct nontrivial solutions; when p and q are real numbers larger than 1, λ=1, Boccardo and Guedes de Figueiredo [17] obtained the existence of solutions of the system (P2).But up to now, to the best of our knowledge, no paper discussing the solutions of problem (P) with nonsmooth potential via nonsmooth critical point theory can be found in the existing literature. In order to fill in this gap, we study problem (P) from a more extensive viewpoint. More precisely, we would study the existence of at least two nontrivial solutions for the problem (P) as the parameter λ>λ0 for some constant λ0.This paper is divided into three sections: in the second section we introduce some necessary knowledge on the nonsmooth analysis, basic properties of the generalized Lebesgue-spaceLp(x)(Ω) and the generalized Lebesgue-Sobolev space W1,p(x)(Ω). In the third section, we give the assumptions on the nonsmooth potential F(x,t,s) and prove the multiplicity results for problem (P).


## 2. Preliminary

### 2.1. Variable Exponent Sobolev Space

In order to discuss problem (P), we need some theories on W0k,p(x)(Ω) which we call variable exponent Sobolev space. Firstly we review some facts on variable exponent spaces Lp(x)(Ω) and Wk,p(x)(Ω). For the details see [4, 18–20].Firstly, we need to give some notations, which we shall use through this paper:(2.1)C+(Ω¯)={p∈C(Ω¯):p(x)>1foranyx∈Ω¯},p-=lim⁡x∈Ω¯⁡p(x),p+=max⁡x∈Ω¯⁡p(x)foranyp∈C+(Ω¯).
Obviously, 1<p-≤p+<+∞.Denote by𝒰(Ω) the set of all measurable real functions defined on Ω. Two functions in 𝒰(Ω) are considered to be one element of 𝒰(Ω), when they are equal almost everywhere.Forp∈C+(Ω¯), define(2.2)Lp(x)(Ω)={u∈U(Ω):∫Ω|u(x)|p(x)dx<+∞},
and with the norm |u|Lp(x)(Ω)=|u|p(x)=inf{λ>0:∫Ω|u(x)/λ|p(x)dx≤1},W1,p(x)(Ω)={u∈Lp(x)(Ω):|∇u|∈Lp(x)(Ω)}, with the norm ∥u∥=∥u∥W1,p(x)(Ω)=|u|p(x)+|∇u|p(x).DenoteW01,p(x)(Ω) as the closure of C0∞(Ω) in W1,p(x)(Ω).Hereafter, let(2.3)p*(x)={Np(x)N-p(x),p(x)<N,+∞,p(x)≥N.Lemma 2.1 (see [19]).
(1) The spacesLp(x)(Ω), W1,p(x)(Ω), and W01,p(x)(Ω) are separable and reflexive Banach spaces. Moreover, Lp(x)(Ω) is uniform convex.
(2) Poincare inequality inW01,p(x)(Ω) holds; that is, there exists a positive constant C such that
(2.4)|u|Lp(x)(Ω)≤C|∇u|Lp(x)(Ω),∀u∈W01,p(x)(Ω).
(3) Ifq∈C+(Ω¯) and q(x)<p*(x) for any x∈Ω¯, then the embedding from W1,p(x)(Ω) to Lq(x)(Ω) is compact and continuous.By (2) of Lemma2.1, we know that |∇u|p(x) and ∥u∥ are equivalent norms on W01,p(x)(Ω). We will use |∇u|p(x) to replace ∥u∥ in the following discussions.Lemma 2.2 (see [4]).
The conjugate space ofLp(x)(Ω) is Lq(x)(Ω), where 1/p(x)+1/q(x)=1. For any u∈Lp(x)(Ω) and v∈Lq(x)(Ω), one has
(2.5)∫Ω|uv|dx≤(1p-+1q-)|u|Lp(x)(Ω)|v|Lq(x)(Ω).Lemma 2.3 (see [4]).
Setρ(u)=∫Ω|u(x)|p(x)dx. For u,uk∈Lp(x)(Ω), one has(1)
foru≠0,|u|p(x)=λ⇔ρ(u/λ)=1;(2)
|u|p(x)<1(=1;>1)⇔ρ(u)<1(=1;>1);(3)
if|u|p(x)>1, then |u|p(x)p-≤ρ(u)≤|u|p(x)p+;(4)
if|u|p(x)<1, then |u|p(x)p+≤ρ(u)≤|u|p(x)p-;(5)
limk→+∞|uk|p(x)=0⇔Limk→+∞ρ(uk)=0;(6)
|uk|p(x)→+∞⇔ρ(uk)→+∞.In this paper, the spaceW01,p(x)(Ω)×W01,q(x)(Ω) will be endowed with the following equivalent norm:(2.6)∥(u,v)∥=∥u∥+∥v∥,
where(2.7)∥u∥=inf⁡⁡{λ>0:∫Ω|∇uλ|p(x)dx≤1},∥v∥=inf⁡{λ>0:∫Ω|∇vλ|q(x)dx≤1}.Similar to Lemma2.3, we have the following.Lemma 2.4.
Setρ(u)=∫Ω|∇u(x)|p(x)dx. For u,uk∈W1,p(x)(Ω), one has(1)
foru≠0,∥u∥=λ⇔ρ(u/λ)=1;(2)
∥u∥<1(=1;>1)⇔ρ(u)<1(=1;>1);(3)
if∥u∥>1, then ∥u∥p-≤ρ(u)≤∥u∥p+;(4)
if∥u∥<1, then ∥u∥p+≤ρ(u)≤∥u∥p-;(5)
Limk→+∞∥uk∥=0⇔Limk→+∞ρ(uk)=0;(6)
∥uk∥→+∞⇔ρ(uk)→+∞.
Consider the following function:(2.8)J(u)=∫Ω1p(x)|∇u|p(x)dx,u∈W01,p(x)(Ω).
We know that (see [21]) J∈C1(W01,p(x)(Ω),R) and p(x)-Laplacian operator -Δp(x)u=-div(|∇u|p(x)-2∇u) is the derivative operator of J in the weak sense. We denote ℒ=J′:W01,p(x)(Ω)→(W01,p(x)(Ω))*, then 〈ℒ(u),v〉=∫Ω(|∇u(x)|p(x)-2∇u·∇vdx,for allu,v∈W01,p(x)(Ω).Lemma 2.5 (see [19]).
SetX=W01,p(x)(Ω), ℒ is as above, then(1)
ℒ:X→X* is a continuous, bounded and strictly monotone operator;(2)
ℒ is a mapping of type (S+), if un⇀u(weak) in X andlimsupn→∞〈ℒ(un),un-u〉≤0, then un→u in X;(3)
ℒ:X→X* is a homeomorphism.


### 2.2. Generalized Gradient

LetX be a Banach space, X*  its topological dual space and we denote 〈·,·〉 as the duality bracket for pair (X*,X). A function φ:X↦ℝ is said to be locally Lipschitz, if for every x∈X we can find a neighbourhood U of x and a constant k>0(depending on U), such that |φ(y)-φ(z)|≤k∥y-z∥,for ally,z∈U.The generalized directional derivative ofφ at the point u∈X in the direction h∈X is(2.9)φ0(u;h)=limsup⁡u′→u;λ↓0φ(u′+λh)-φ(u′)λ.The generalized subdifferential ofφ at the point u∈X is defined by the(2.10)∂φ(u)={u*∈X*;〈u*,h〉≤φ0(u;h),∀h∈X},
which is a nonempty, convex and w*-compact set of X. We say that u∈X is a critical point of φ, if 0∈∂φ(u). For further details, we refer the reader to [12].Finally we have the following Weierstrass Theorem and Mountain Pass Theorem.Theorem 2.6.
IfX is a reflexive Banach space and φ:X→ℝ satisfies(1)
φ is weak lower semicontinuous, that is,
(2.11)xn⇀x0(weakly)in⁡X⟶φ(x0)≤lim⁡inf⁡n→+∞φ(xn);(2)
φ is coercive, that is, lim∥x∥→∞φ(x)=+∞,then there exists x*∈X such that φ(x*)=minx∈Xφ(x).Theorem 2.7 (see [22]).
Letφ:X→ℝ be locally Lipschitz function and x0,x1∈X. If there exists a bounded open neighbourhood U of x0, such that x1∈X∖U, max{φ(x0),φ(x1)}<inf∂Uφ and φ satisfies the nonsmooth C-condition at level c, where c=infγ∈𝒯maxt∈[0,1]φ(γ(t)),𝒯={γ∈C([0,1];X):γ(0)=x0,γ(1)=x1}, then c is a critical value of φ and c≥inf∂Uφ.


## 2.1. Variable Exponent Sobolev Space

In order to discuss problem (P), we need some theories on W0k,p(x)(Ω) which we call variable exponent Sobolev space. Firstly we review some facts on variable exponent spaces Lp(x)(Ω) and Wk,p(x)(Ω). For the details see [4, 18–20].Firstly, we need to give some notations, which we shall use through this paper:(2.1)C+(Ω¯)={p∈C(Ω¯):p(x)>1foranyx∈Ω¯},p-=lim⁡x∈Ω¯⁡p(x),p+=max⁡x∈Ω¯⁡p(x)foranyp∈C+(Ω¯).
Obviously, 1<p-≤p+<+∞.Denote by𝒰(Ω) the set of all measurable real functions defined on Ω. Two functions in 𝒰(Ω) are considered to be one element of 𝒰(Ω), when they are equal almost everywhere.Forp∈C+(Ω¯), define(2.2)Lp(x)(Ω)={u∈U(Ω):∫Ω|u(x)|p(x)dx<+∞},
and with the norm |u|Lp(x)(Ω)=|u|p(x)=inf{λ>0:∫Ω|u(x)/λ|p(x)dx≤1},W1,p(x)(Ω)={u∈Lp(x)(Ω):|∇u|∈Lp(x)(Ω)}, with the norm ∥u∥=∥u∥W1,p(x)(Ω)=|u|p(x)+|∇u|p(x).DenoteW01,p(x)(Ω) as the closure of C0∞(Ω) in W1,p(x)(Ω).Hereafter, let(2.3)p*(x)={Np(x)N-p(x),p(x)<N,+∞,p(x)≥N.Lemma 2.1 (see [19]).
(1) The spacesLp(x)(Ω), W1,p(x)(Ω), and W01,p(x)(Ω) are separable and reflexive Banach spaces. Moreover, Lp(x)(Ω) is uniform convex.
(2) Poincare inequality inW01,p(x)(Ω) holds; that is, there exists a positive constant C such that
(2.4)|u|Lp(x)(Ω)≤C|∇u|Lp(x)(Ω),∀u∈W01,p(x)(Ω).
(3) Ifq∈C+(Ω¯) and q(x)<p*(x) for any x∈Ω¯, then the embedding from W1,p(x)(Ω) to Lq(x)(Ω) is compact and continuous.By (2) of Lemma2.1, we know that |∇u|p(x) and ∥u∥ are equivalent norms on W01,p(x)(Ω). We will use |∇u|p(x) to replace ∥u∥ in the following discussions.Lemma 2.2 (see [4]).
The conjugate space ofLp(x)(Ω) is Lq(x)(Ω), where 1/p(x)+1/q(x)=1. For any u∈Lp(x)(Ω) and v∈Lq(x)(Ω), one has
(2.5)∫Ω|uv|dx≤(1p-+1q-)|u|Lp(x)(Ω)|v|Lq(x)(Ω).Lemma 2.3 (see [4]).
Setρ(u)=∫Ω|u(x)|p(x)dx. For u,uk∈Lp(x)(Ω), one has(1)
foru≠0,|u|p(x)=λ⇔ρ(u/λ)=1;(2)
|u|p(x)<1(=1;>1)⇔ρ(u)<1(=1;>1);(3)
if|u|p(x)>1, then |u|p(x)p-≤ρ(u)≤|u|p(x)p+;(4)
if|u|p(x)<1, then |u|p(x)p+≤ρ(u)≤|u|p(x)p-;(5)
limk→+∞|uk|p(x)=0⇔Limk→+∞ρ(uk)=0;(6)
|uk|p(x)→+∞⇔ρ(uk)→+∞.In this paper, the spaceW01,p(x)(Ω)×W01,q(x)(Ω) will be endowed with the following equivalent norm:(2.6)∥(u,v)∥=∥u∥+∥v∥,
where(2.7)∥u∥=inf⁡⁡{λ>0:∫Ω|∇uλ|p(x)dx≤1},∥v∥=inf⁡{λ>0:∫Ω|∇vλ|q(x)dx≤1}.Similar to Lemma2.3, we have the following.Lemma 2.4.
Setρ(u)=∫Ω|∇u(x)|p(x)dx. For u,uk∈W1,p(x)(Ω), one has(1)
foru≠0,∥u∥=λ⇔ρ(u/λ)=1;(2)
∥u∥<1(=1;>1)⇔ρ(u)<1(=1;>1);(3)
if∥u∥>1, then ∥u∥p-≤ρ(u)≤∥u∥p+;(4)
if∥u∥<1, then ∥u∥p+≤ρ(u)≤∥u∥p-;(5)
Limk→+∞∥uk∥=0⇔Limk→+∞ρ(uk)=0;(6)
∥uk∥→+∞⇔ρ(uk)→+∞.
Consider the following function:(2.8)J(u)=∫Ω1p(x)|∇u|p(x)dx,u∈W01,p(x)(Ω).
We know that (see [21]) J∈C1(W01,p(x)(Ω),R) and p(x)-Laplacian operator -Δp(x)u=-div(|∇u|p(x)-2∇u) is the derivative operator of J in the weak sense. We denote ℒ=J′:W01,p(x)(Ω)→(W01,p(x)(Ω))*, then 〈ℒ(u),v〉=∫Ω(|∇u(x)|p(x)-2∇u·∇vdx,for allu,v∈W01,p(x)(Ω).Lemma 2.5 (see [19]).
SetX=W01,p(x)(Ω), ℒ is as above, then(1)
ℒ:X→X* is a continuous, bounded and strictly monotone operator;(2)
ℒ is a mapping of type (S+), if un⇀u(weak) in X andlimsupn→∞〈ℒ(un),un-u〉≤0, then un→u in X;(3)
ℒ:X→X* is a homeomorphism.


## 2.2. Generalized Gradient

LetX be a Banach space, X*  its topological dual space and we denote 〈·,·〉 as the duality bracket for pair (X*,X). A function φ:X↦ℝ is said to be locally Lipschitz, if for every x∈X we can find a neighbourhood U of x and a constant k>0(depending on U), such that |φ(y)-φ(z)|≤k∥y-z∥,for ally,z∈U.The generalized directional derivative ofφ at the point u∈X in the direction h∈X is(2.9)φ0(u;h)=limsup⁡u′→u;λ↓0φ(u′+λh)-φ(u′)λ.The generalized subdifferential ofφ at the point u∈X is defined by the(2.10)∂φ(u)={u*∈X*;〈u*,h〉≤φ0(u;h),∀h∈X},
which is a nonempty, convex and w*-compact set of X. We say that u∈X is a critical point of φ, if 0∈∂φ(u). For further details, we refer the reader to [12].Finally we have the following Weierstrass Theorem and Mountain Pass Theorem.Theorem 2.6.
IfX is a reflexive Banach space and φ:X→ℝ satisfies(1)
φ is weak lower semicontinuous, that is,
(2.11)xn⇀x0(weakly)in⁡X⟶φ(x0)≤lim⁡inf⁡n→+∞φ(xn);(2)
φ is coercive, that is, lim∥x∥→∞φ(x)=+∞,then there exists x*∈X such that φ(x*)=minx∈Xφ(x).Theorem 2.7 (see [22]).
Letφ:X→ℝ be locally Lipschitz function and x0,x1∈X. If there exists a bounded open neighbourhood U of x0, such that x1∈X∖U, max{φ(x0),φ(x1)}<inf∂Uφ and φ satisfies the nonsmooth C-condition at level c, where c=infγ∈𝒯maxt∈[0,1]φ(γ(t)),𝒯={γ∈C([0,1];X):γ(0)=x0,γ(1)=x1}, then c is a critical value of φ and c≥inf∂Uφ.


## 3. Existence Results

For each(u,v)∈W01,p(x)(Ω)×W01,q(x)(Ω), define(3.1)Φ(u,v)=∫Ω1p(x)|∇u|p(x)dx+∫Ω1p(x)|∇v|q(x)dx,Ψ(u,v)=∫ΩF(x,u,v)dx.By a solution of (2), we mean function (u,v)∈W01,p(x)(Ω)×W01,q(x)(Ω) to which there corresponds mapping Ω∋x→(w1,w2) with w1(x)∈∂uF(x,u,v), w2(x)∈∂vF(x,u,v) for almost every x∈Ω having the property that for every (ξ,η)∈W01,p(x)(Ω)×W01,q(x)(Ω), the function x→(w1(x)ξ(x),w2(x)η(x))∈L1(Ω)×L1(Ω) and(3.2)∫Ω|∇u|p(x)-2∇u∇ξdx+∫Ω|∇v|q(x)-2∇v∇ηdx=λ∫Ωw1ξdx+λ∫Ωw2ηdx.Our hypotheses on nonsmooth potentialF(x,t,s) is as follows.H(F):F:Ω×ℝ×ℝ→ℝ is a function such that F(x,0,0)=0 a.e. on Ω and satisfies the following facts:(1)
for allt∈ℝ, s∈ℝ, x↦F(x,t,s) is measurable;(2)
for almost allx∈Ω, t↦F(x,t,s) and s↦F(x,t,s) are locally Lipschitz.Lemma 3.1.
SupposeH(F) and the following conditions hold:(f1)
there existsα∈C+(Ω¯) and α(x)<p*(x), such that
(3.3)|w1|≤c1(1+|t|α(x)-1+|s|β(x)(α(x)-1)/α(x)),
for almost all x∈Ω, all t,s∈ℝ and w1∈∂tF(x,t,s);(f2)
there existsβ∈C+(Ω¯) and β(x)<q*(x), such that
(3.4)|w2|≤c2(1+|t|(α(x)(β(x)-1))/(β(x))+|s|β(x)-1),
for almost all x∈Ω, all t,s∈ℝ and w2∈∂sF(x,t,s);
thenφ(·,v)(φ(u,·)) is locally Lipschitz on W01,p(x)(Ω)(W01,q(x)(Ω)).Proof.
We need only to prove thatφ(·,v) is locally Lipschitz on W01,p(x)(Ω).
ByΦ(·,v)∈C1(W01,p(x)(Ω),ℝ),we have
(3.5)φ(u1,v)-φ(u2,v)=J(u1)-J(u2)=J′(u¯)⋅(u1-u2),
whereu¯=tu1+(1-t)u2,t∈(0,1).
LetBr={u∈W01,p(x)(Ω):∥u-u0∥W01,p(x)(Ω)≤r}.
Note thatBr is w-compact. Then we obtain that there exists a positive constant M, such that ∥J′(u¯)∥W-1,p′(x)(Ω)≤M with 1/p(x)+1/p′(x)=1, for sufficiently small r.
Therefore, for anyu1,u2∈Br, we have
(3.6)|Φ(u1,v)-Φ(u2,v)|=|J′(u¯)⋅(u1-u2)|≤‖J′(u¯)‖W-1,p′(x)(x)(Ω)‖u1-u2‖W01,p(x)(Ω)≤M‖u1-u2‖W01,p(x)(Ω),∀v∈W01,q(x)(Ω).
Fixingv∈W01,q(x)(Ω), by (f1) and Lebourg mean value theorem, we have
(3.7)|F(x,u1,v)-F(x,u2,v)|≤c1(1+|u¯|α(x)-1+|v|β(x)(α(x)-1)/α(x))|u1-u2|.
Hence,
(3.8)|∫ΩF(x,u1,v)dx-∫ΩF(x,u2,v)dx|≤c1∫Ω|u1-u2|dx+c1∫Ω|u¯|α(x)-1|u1-u2|dx+c1∫Ω|v|β(x)(α(x)-1)/α(x)|u1-u2|dx≤c¯1|u1-u2|α(x)+c1||u¯|α(x)-1|α′(x)|u1-u2|α(x)+c1||v|β(x)(α(x)-1)/α(x)|α′(x)|u1-u2|α(x),
where1/α′(x)+1/α(x)=1.
It is immediate that(3.9)∫Ω(|u¯|α(x)-1)α′(x)=∫Ω|u¯|α(x)dx≤{|u¯|α(x)α+≤c‖u¯‖α+,|u¯|α(x)>1,|u¯|α(x)α+≤c‖u¯‖α-,|u¯|α(x)<1,∫Ω(|v|β(x)(α(x)-1)/α(x))α′(x)=∫Ω|v|β(x)dx≤{|v|β(x)β+≤c‖v‖β+,|v|β(x)>1,|v|β(x)β+≤c‖v‖β-,|v|β(x)<1
are bounded.
So,(3.10)|∫ΩF(x,u1,v)dx-∫ΩF(x,u2,v)dx|≤c∥u1-u2∥,
since W01,p(x)(Ω)↪Lα(x)(Ω) is a compact embedding.
Therefore,φ(·,v) is locally Lipschitz. Similarly, we can prove that φ(u,·) is locally Lipschitz.Theorem 3.2.
Suppose thatH(F), (f1) with α+<p-, (f2) with β+<q- and the following conditions (f3)-(f4) hold:(f3)
there existsγi∈C(Ω¯)(i=1,2) with p+<γ1(x)<p*(x),q+<γ2(x)<q*(x) and μ1,μ2∈L∞(Ω), such that
(3.11)limsup⁡t→0,s→0⁡〈w1,t〉|t|γ1(x)<μ1(x),limsup⁡t→0,s→0⁡〈w2,s〉|s|γ2(x)<μ2(x)
uniformly for almost all x∈Ω and all w1∈∂tF(x,t,s),w2∈∂sF(x,t,s);(f4)
there existξ0∈R,η0∈R,x0∈Ω and r0>0, such that
(3.12)F(x,ξ0,η0)>δ0>0,a.e.x∈Br0(x0),
where Br0(x0):={x∈Ω:|x-x0|≤r0}⊂Ω.Then there existsλ*>0 such that, for each λ>λ*, the problem (2) has at least two nontrivial solutions.Proof.
The proof is divided into four steps as follows.Step 1.
We will show thatφ is coercive in the step.
Firstly, for almost allx∈Ω, by H(F)(2), t↦F(x,t,s) is differentiable almost everywhere on ℝ and we have
(3.13)ddtF(x,t,s)∈∂tF(x,t,s).
Moreover, from (f1), (f2) and Young inequality, we can get that
(3.14)F(x,t,s)=F(x,0,s)+∫0tddyF(x,y,s)dy=F(x,0,0)+∫0sddzF(x,0,z)dz+∫0tddyF(x,y,s)dy≤c1[|t|+|t|α(x)α(x)+|s|β(x)(α(x)-1)/α(x)|t|]+c2[|s|+|t|α(x)(β(x)-1)/β(x)|s|+1β(x)|s|β(x)]≤c1[|t|+|t|α(x)+(α(x)-1)|s|β(x)α(x)+|t|α(x)β(x)]+c2[|s|+|s|β(x)+(β(x)-1)|t|α(x)β(x)+|s|β(x)β(x)]≤c1[|t|+2|t|α(x)+|s|β(x)]+c2[|s|+2|s|β(x)+|t|α(x)],
for almost all x∈Ω and t,s∈ℝ.
Note that1<α(x)≤α+<p-<p*(x) and 1<β(x)≤β+<q-<q*(x), then, by Lemma 2.1, we have W01,p(x)(Ω)↪Lα(x)(Ω) and W01,q(x)(Ω)↪Lβ(x)(Ω) (compact embedding). Furthermore, there exists c3,c4>0 such that |u|α(x)≤c3∥u∥ and |v|β(x)≤c4∥v∥.
So, for any|u|α(x)>1 and ∥u∥>1,
(3.15)∫Ω|u|α(x)dx≤|u|α(x)α+≤c3α+‖u‖α+
and, for any |v|β(x)>1 and ∥v∥>1,
(3.16)∫Ω|v|β(x)dx≤|v|β(x)β+≤c4β+‖v‖β+.
By (3.14), (3.15), (3.31), the Hölder inequality and the Sobolev embedding theorem, we have
(3.17)φ(u,v)=∫Ω1p(x)|∇u|p(x)dx+∫Ω1q(x)|∇v|q(x)dx-λ∫ΩF(x,u,v)dx≥1p+‖u‖p-+1q+‖v‖q--λc1∫Ω|u|dx-2c1c3α+‖u‖α+-c1c4β+‖v‖β+-λc2∫Ω|v|dx-2c2c4β+‖v‖β+-c2c3α+‖u‖α+≥1p+‖u‖p-+1q+‖v‖q--2λc1|1|α′(x)|u|α(x)-2λc1|1|β′(x)|v|β(x)-2c1c3α+‖u‖α+-c1c4β+‖v‖β+-2c2c4β+‖v‖β+-c2c3α+‖u‖α+≥1p+‖u‖p-+1q+‖v‖q--2λc1c|1|α′(x)‖u‖-2λc1c|1|β′(x)‖v‖-2c1c3α+‖u‖α+-c1c4β+‖v‖β+-2c2c4β+‖v‖β+-c2c3α+‖u‖α+⟶∞,as‖u,v‖⟶∞.Step 2.
We will show that theφ is weakly lower semicontinuous.
Letingun⇀u weakly in W01,p(x)(Ω), vn⇀v weakly in W01,q(x)(Ω) by Lemma 2.1(3), we obtain the following results:W01,p(x)(Ω)↪Lp(x)(Ω);
W01,q(x)(Ω)↪Lq(x)(Ω);un→u in Lp(x)(Ω);vn→v in Lq(x)(Ω);un→u for a.a. x∈Ω;vn→v for a.a. x∈Ω;F(x,un(x),vn(x))→F(x,u(x),v(x)) for a.a. x∈Ω.
By Fatou’s Lemma,(3.18)limsup⁡n→∞⁡∫ΩF(x,un(x),vn(x))dx≤∫ΩF(x,u(x),v(x))dx.
Thus,(3.19)lim⁡inf⁡n→∞⁡φ(un,vv)=lim⁡inf⁡n→∞⁡[∫Ω1p(x)|∇un|p(x)dx+∫Ω1q(x)|∇vn|q(x)dx]-limsup⁡n→∞⁡∫ΩF(x,un,vn)dx≥∫Ω1p(x)|∇u|p(x)dx+∫Ω1q(x)|∇v|q(x)dx-λ∫ΩF(x,u,v)dx.=φ(u)
Hence, by Theorem2.6, we deduce that there exists a global minimizer (u0,v0)∈W01,p(x)(Ω)×W01,q(x)(Ω) such that
(3.20)φ(u0,v0)=min⁡(u,v)∈W01,p(x)(Ω)×W01,q(x)(Ω)φ(u,v).Step 3.
We will show that there existsλ*>0 such that for each λ>λ*, φ(u0,v0)<0.
By the condition (f4), there exists ξ0,η0∈ℝ such that F(x,ξ0,η0)>δ0>0, a.e. x∈Br0(x0). It is clear that
(3.21)0<M1:=max⁡|t|≤|ξ0|,|s|≤|η0|⁡{[|t|+2|t|α-+|s|β-]+c2[|s|+2|s|β-+|t|α-]c1[|t|+2|t|α++|s|β+]+c2[|s|+2|s|β++|t|α+],c1[|t|+2|t|α-+|s|β-]+c2[|s|+2|s|β-+|t|α-]c1[|t|+2|t|α++|s|β+]+c2[|s|+2|s|β++|t|α+]}<+∞.
Now we denote(3.22)t0=(M1δ0+M1)1/N,K(t):=max⁡{(ξ0r0(1-t))p-,(ξ0r0(1-t))p+,(ξ0r0(1-t))q-,(ξ0r0(1-t))q+},λ*=max⁡t∈[t1,t2]K(t)(1-tN)[δ0tN-M1(1-tN)],
where t0<t1<t2<1 and δ0 is given in the condition (f4). A simple calculation shows that the function t↦δ0tN-M1(1-tN) is positive whenever t>t0 and δ0t0N-M1(1-t0N)=0. Thus λ* is well defined and λ*>0.
We will show that, for eachλ>λ*, the problem (P) has two nontrivial solutions. In order to do this, for t∈[t1,t2], let us define
(3.23)ηt(x)={0,ifx∈Ω∖Br0(x0),ξ0,ifx∈Btr0(x0),ξ0r0(1-t)(r0-|x-x0|),ifx∈Br0(x0)∖Btr0(x0).
By conditions (f1) and (f3) we have
(3.24)∫ΩF(x,ηt(x),ηt(x))dx=∫Btr0(x0)F(x,ηt(x),ηt(x))dx+∫Br0(x0)∖Btr0(x0)F(x,ηt(x),ηt(x))dx≥wNr0NtNδ0-M1(1-tN)wNr0N=wNr0N(δ0tN-M1(1-tN)).
Hence, for t∈[t1,t2],
(3.25)φ(ηt,ηt)=∫Ω1p(x)|∇ηt|p(x)dx+∫Ω1q(x)|∇ηt|q(x)dx-λ∫ΩF(x,ηt(x),ηt(x))dx≤1p-∫Ω|∇ηt|p(x)dx+1q-∫Ω|∇ηt|q(x)dx-λwNr0N(δ0tN-M1(1-tN))≤max⁡{(ξ0r0(1-t))p-,(ξ0r0(1-t))p+,(ξ0r0(1-t))q-,(ξ0r0(1-t))q+}×wNr0N(1-tN)-λwNr0N(δ0tN-M1(1-tN))=wNr0N[K(t)(1-tN)-λ(δ0tN-M1(1-tN))],
so that φ(ηt,ηt)<0 whenever λ>λ*.Step 4.
We will check the C-condition in the following.
Suppose{(un,vn)}n≥1⊆W01,p(x)(Ω)×W01,q(x)(Ω) such that φ(un,vn)→c and (1+∥(un,vn)∥)m(un,vn)→0. Let (un*,vn*)∈∂φ(un,vn) be such that m(un,vn)=∥(un*,vn*)∥(W01,p(x)(Ω)×W01,q(x)(Ω))*. The interpretation of (un*,vn*)∈∂φ(un,vn)is that un*∈∂uφ(un,vn) and vn*∈∂vφ(un,vn). We know that
(3.26)un*=-Δp(x)un-λwn1,vn*=-Δq(x)vn-λwn2
with wn1∈∂uΨ(un,vn) and wn2∈∂vΨ(un,vn). From Chang [23] we know that wn1∈Lα′(x)(Ω) and wn2∈Lβ′(x)(Ω), where α′(x)=α(x)/(α(x)-1),β′(x)=β(x)/(β(x)-1).
Sinceφ is coercive, {un}n≥1, {vn}n≥1 are bounded and passed to a subsequence, still denoting{un}n≥1 and {vn}n≥1, we may assume that there exist u∈W01,p(x)(Ω), v∈W01,q(x)(Ω), such that un⇀u weakly in W01,p(x)(Ω) and vn⇀v weakly in W01,q(x)(Ω). Next we will prove that
(3.27)un→uinW01,p(x)(Ω),vn→vinW01,q(x)(Ω).
ByW01,p(x)(Ω)→Lp(x)(Ω), W01,q(x)(Ω)→Lq(x)(Ω), we have un→u in Lp(x)(Ω) and vn→v in Lq(x)(Ω). Moreover, since ∥(un*,vn*)∥(W01,p(x)(Ω)×W01,q(x)(Ω))*→0, we get ∥un*∥(W01,p(x)(Ω))*→0, ∥vn*∥(W01,q(x)(Ω))*→0, so |〈un*,un〉|≤εn, |〈vn*,vn〉|≤εn.
From (3.26), we have
(3.28)〈-Δp(x)un,un-u〉-∫Ωwn1(un-u)dx≤εn,∀n≥1.〈-Δq(x)vn,vn-v〉-∫Ωwn2(vn-v)dx≤εn,∀n≥1.
Moreover, ∫Ωwn1(un-u)dx→0 and ∫Ωwn2(vn-v)dx→0, since un→u in Lp(x)(Ω), vn→v in Lq(x)(Ω), {wn1}n≥1 in Lp′(x)(Ω) and {wn2}n≥1 in Lq′(x)(Ω) are bounded, where 1/p(x)+1/(p′(x))=1,1/(q(x))+1/(q′(x))=1. Therefore,
(3.29)limsup⁡n→∞〈-Δp(x)un,un-u〉≤0,limsup⁡n→∞〈-Δq(x)vn,vn-v〉≤0.
From Lemma2.5, we have un→u, vn→v as n→∞. Thus φ satisfies the nonsmooth C-condition.Step 5.
We will show that there exists another nontrivial weak solution of problem (P).
From Lebourg Mean Value Theorem, we obtain(3.30)F(x,t,s)-F(x,0,s)=〈w1,t〉,F(x,0,s)-F(x,0,0)=〈w2,s〉
for some w1∈∂tF(x,ϑt,s), w2∈∂sF(x,0,τs), and 0<ϑ,τ<1. Thus by the condition (f3), there exists β∈(0,1) such that
(3.31)|F(x,t,s)|≤|〈w1,t〉|+|〈w2,s〉|≤μ1(x)|t|γ1(x)+μ2(x)|s|γ2(x),
for all |t|,|s|<βand a.e.x∈Ω.
It follows from the conditions (f1), (f2) 1<α-≤α+<p-≤p+<γ1(x)<p*(x) and 1<β-≤β+<q-≤q+<γ2(x)<p*(x) that for all |t|>β, |s|>β and a.e. x∈Ω,
(3.32)|F(x,t,s)|≤c1[|t|+2|t|α(x)+|s|β(x)]+c2[|s|+2|s|β(x)+|t|α(x)]≤(c1βγ1(x)-1+2c2βγ1(x)-α(x)+c2βγ1(x)-β(x))|t|γ1(x)+(c2βγ2(x)-1+2c2βγ2(x)-β(x)+c1βγ1(x)-α(x))|t|γ2(x)≤(c1βγ1+-1+2c2βγ1+-α-+c2βγ1+-β-)|t|γ1(x)+(c2βγ2+-1+2c2βγ2+-β-+c1βγ1+-α-)|t|γ2(x),
this together with (3.31) yields that, for all t∈ℝ and a.e. x∈Ω,
(3.33)|F(x,t,s)|≤(μ1(x)+c1βγ1+-1+2c2βγ1+-α-+c2βγ1+-β-)|t|γ1(x)+(μ2(x)+c2βγ2+-1+2c2βγ2+-β-+c1βγ1+-α-)|t|γ2(x)≤c3|t|γ1(x)+c4|s|γ2(x),
for positive constants c3,c4.
Note thatp+<γ1(x)<p*(x), q+<γ2(x)<q*(x), then, by Lemma 2.1, we have W01,p(x)(Ω)↪Lγ1(x)(Ω) and W01,q(x)(Ω)↪Lγ2(x)(Ω). Furthermore, there exist c5,c6 such that
(3.34)|u|γ1(x)≤c5‖u‖,∀u∈W01,p(x)(Ω),|v|γ2(x)≤c6‖u‖,∀v∈W01,q(x)(Ω).
For allλ>λ*, ∥(u,v)∥<1, |u|γ1(x)<1 and |v|γ2(x)<1, from (3.33) we have
(3.35)φ(u,v)=∫Ω1p(x)|∇u|p(x)dx+∫Ω1q(x)|∇v|q(x)dx-λ∫ΩF(x,u(x),v(x))dx≥∫Ω1p(x)|∇u|p(x)dx+∫Ω1q(x)|∇v|q(x)dx-λc3∫Ω|u(x)|γ1(x)dx-λc4∫Ω|v(x)|γ2(x)dx≥1p+‖u‖p++1q+‖v‖q+-λ[c3c5γ1-‖u‖γ1-+c4c6γ2-‖u‖γ2-].
So, forρ>0 small enough, there exists a ν>0 such that
(3.36)φ(u,v)>ν,for‖u,v‖=ρ
and ∥(u0,v0)∥>ρ. So by the Nonsmooth Mountain Pass Theorem (cf. Theorem 2.7), we can get (u1,v1)∈W01,p(x)(Ω)×W01,q(x)(Ω)  which satisfies
(3.37)φ(u1,v1)=c>0,m(u1,v1)=0.
Therefore,(u1,v1) is another nontrivial solution of problem (P).Remark 3.3.
Letp->α+, q->β+ and consider the following nonsmooth locally Lipschitz function:
(3.38)F(x,t,s)={tγ1(x)+sγ2(x),t∈(0,δ)ors∈(0,δ),max⁡⁡{|t-δ|θ(x),|t-δ|α(x)}+|δ|γ1(x)+max⁡{|s-δ|θ(x),|s-δ|β(x)}+|δ|γ2(x),t∈[δ,+∞)ors∈[δ,+∞),0,t∈(-∞,0]ors∈(-∞,0],
where 0<δ<1, θ->1, θ+<α- and θ+<β-.
Obvious,t↦F(x,t,s) and s↦F(x,t,s) are locally Lipschitz. Then,
(3.39)∂tF(x,t,s)={γ1(x)tγ1(x)-1,t∈(0,δ),θ(x)(t-δ)θ(x)-1,t∈(δ,1+δ),α(x)(t-δ)α(x)-1,t∈(1+δ,+∞),[0,γ1(x)δγ1(x)-1],t=δ,[θ(x),α(x)],t=1+δ,0,t∈(-∞,0],∂sF(x,t,s)={γ2(x)sγ2(x)-1,s∈(0,δ),θ(x)(s-δ)θ(x)-1,s∈(δ,1+δ),β(x)(t-δ)β(x)-1,s∈(1+δ,+∞),[0,γ1(x)δγ1(x)-1],s=δ,[θ(x),β(x)],s=1+δ,0,s∈(-∞,0].
Hence, for anyw1∈∂tF(x,t,s) and w2∈∂sF(x,t,s), we have
(3.40)|w1|≤{γ1(x)tα(x)-1tγ1(x)-α(x)≤γ1+|t|α(x)-1,t∈(0,δ),θ(x)(t-δ)θ(x)-1<θ+<θ+(1δ)α+-1|t|α(x)-1,t∈(δ,1+δ),α(x)(t-δ)α(x)-1≤α+|t|α(x)-1,t∈(1+δ,+∞),γ1(x)δγ1(x)-1=γ1(x)δα(x)-1δγ1(x)-α(x)≤γ+δα(x)-1,t=δ,[θ(x),α(x)]≤α+(1+δ)α(x)-1,t=1+δ,0,t∈(-∞,0],|w2|≤{γ2(x)sβ(x)-1sγ2(x)-β(x)≤γ2+|s|β(x)-1,s∈(0,δ),θ(x)(s-δ)θ(x)-1<θ+<θ+(1δ)β+-1|s|β(x)-1,s∈(δ,1+δ),β(x)(s-δ)β(x)-1≤β+|s|β(x)-1,s∈(1+δ,+∞),γ2(x)δγ2(x)-1=γ2(x)δβ(x)-1δγ2(x)-β(x)≤γ+δβ(x)-1,s=δ,[θ(x),β(x)]≤β+(1+δ)β(x)-1,s=1+δ,0,s∈(-∞,0].
Therefore,(3.41)|w1|≤(γ1++α++θ+(1δ)(α+-1))|t|α(x)-1,∀w1∈∂tF(x,t,s),|w2|≤(γ2++β++θ+(1δ)(β+-1))|t|β(x)-1,∀w2∈∂sF(x,t,s),limsup⁡t→0,s→0<w1,t>|t|γ1(x)={lim⁡t→0γ1(x)tγ1(x)tγ1(x),t>00,t≤0}≤γ1(x),limsup⁡t→0,s→0<w2,s>|s|γ2(x)={lim⁡s→0γ2(x)sγ2(x)sγ2(x),s>00,s≤0}≤γ2(x),
uniformly for almost all x∈Ω, all w1∈∂tF(x,t,s) and w2∈∂sF(x,t,s).
Thus far the results involved potential functions exhibitingp(x)-sublinear. The next theorem concerns problems where the potential function is p(x)-superlinear.Theorem 3.4.
Suppose that H(F), (f1) with α->p+, (f2) with β->q+, (f3), (f4) and the following condition (f5) hold. (f5)
For almost allx∈Ω and all t,s∈ℝ, one has F(x,t,s)≤κ(x)withκ∈Lγ3(x)(Ω),1≤γ3(x)<min{p-,q-}.
Then there exists aλ*>0 such that, for each λ>λ*, the problem (P) has at least two nontrivial solutions.Proof.
The steps are similar to those of Theorem3.2. In fact,we only need to modify Step 1 and Step 5 as follows: Step 6 shows, that φ is coercive under the condition (f5); Step 7 shows, that there exists second nontrivial solution under the conditions (f1), (f2), and (f3). Then from Steps 6, 2, 3, 4, and 7 above, the problem (P) has at least two nontrivial solutions.Step 6.
By (f5), for all (u,v)∈W01,p(x)(Ω)×W01,q(x)(Ω), ∥(u,v)∥>1, we have
(3.42)φ(u,v)=∫Ω1p(x)|∇u|p(x)dx+∫Ω1q(x)|∇v|q(x)dx-λ∫ΩF(x,u(x),v(x))dx≥1p+‖u‖p++1q+‖v‖q+-λ∫Ωκ(x)dx⟶∞⟶∞,as‖(u,v)‖→∞.Step 7.
Because of hypothesis (f1), (f2) and mean value theorem for locally Lipschitz functions, we have
(3.43)F(x,t,s)≤c1[|t|+2|t|α(x)+|s|β(x)]+c2[|s|+2|s|β(x)+|t|α(x)]≤c1[|tβ|α(x)-1|t|+2|t|α(x)+|s|β(x)]+c2[|sβ|β(x)-1|s|+2|s|β(x)+|t|α(x)]≤c1[|1β|α+-1|t|α(x)+2|t|α(x)+|s|β(x)]+c2[|1β|β+-1|t|β(x)+2|s|β(x)+|t|α(x)]=c7|t|α(x)+c8|t|β(x)
for a.e. x∈Ω, all |t|≥β, |s|≥β with c7,c8>0.
Combining (3.31) and (3.43), it follows that
(3.44)|F(x,t,s)|≤[μ1(x)|t|γ1(x)+c7|t|α(x)]+[μ2(x)|s|γ2(x)+c8|s|α(x)]
for a.e. x∈Ω and all t,s∈ℝ.
Thus, for allλ>λ*, ∥(u,v)∥<1, |u|γ1(x)<1, |u|α(x)<1|v|γ2(x)<1 and |v|β(x)<1, we have
(3.45)φ(u,v)=∫Ω1p(x)|∇u|p(x)dx+∫Ω1q(x)|∇v|q(x)dx-λ∫ΩF(x,u(x).v(x))dx≥1p+‖u‖p++1q+‖v‖q+-λ∫Ωμ1(x)|u|γ1(x)dx-λc7∫Ω|u|α(x)dx-λ∫Ωμ2(x)|v|γ2(x)dx-λc8∫Ω|v|β(x)dx≥1p+‖u‖p++1q+‖v‖q+-λc9‖u‖γ1--λc7‖u‖α--λc10‖v‖γ2--λc8‖v‖β-.
So, forρ>0 small enough, there exists a ν>0 such that
(3.46)φ(u,v)>ν,for‖u,v‖=ρ
and ∥(u0,v0)∥>ρ. Arguing as in proof of Step 4 of Theorem 3.2, we conclude that φ satisfies the nonsmooth C-condition. So by the Nonsmooth Mountain Pass Theorem (cf. Theorem 2.7), we can get (u1,v1)∈W01,p(x)(Ω)×W01,q(x)(Ω)  which satisfies
(3.47)φ(u1,v1)=c>0,m(u1,v1)=0.
Therefore,(u1,v1) is second nontrivial of problem (P).Remark 3.5.
Consider the following nonsmooth locally Lipschitz function:(3.48)F(x,t,s)={-|t|γ1(x)-|s|γ2(x),|t|≤1or|s|≤1,cos⁡⁡(π|t|)+cos⁡⁡(π|s|),|t|>1;or|s|>1,
In the following, we will show that F(x,t,s) satisfies hypotheses H(F) and (f1)–(f5).
It is clear thatF(x,0,0)=0 for a.e. x∈Ω,thenhypotheses H(F) is satisfied. A direct verification shows that conditions (f4) and (f5) are satisfied. Note that
(3.49)∂tF(x,t,s)={-γ1(x)tγ1(x)-1,0≤t<1,γ1(x)(-t)γ1(x)-1,-1<t≤0,[-γ1(x),0],t=1,[0,γ1(x)],t=-1,{-πsin⁡(πt)},|t|>1,∂sF(x,t,s)={-γ2(x)sγ2(x)-1,0≤s<1,γ2(x)(-s)γ1(x)-1,-1<s≤0,[-γ2(x),0],s=1,[0,γ2(x)],s=-1,{-πsin⁡(πs)},|s|>1.
So,(3.50)|w1|≤(γ1(x)+π)|t|γ1(x)-1,∀w1∈∂tF(x,t,s),|w1|≤(γ2(x)+π)|s|γ2(x)-1,∀w2∈∂sF(x,t,s),limsup⁡t→0,s→0<w1,t>|t|γ1(x)={lim⁡t→0-γ1(x)tγ1(x)tγ1(x),t>0lim⁡t→0-γ1(x)(-t)γ1(x)(-t)γ1(x),t≤0}=-γ1(x)≤γ1(x),limsup⁡t→0,s→0<w2,s>|s|γ2(x)={lim⁡t→0-γ2(x)sγ2(x)sγ2(x),s>0lim⁡t→0-γ2(x)(-s)γ2(x)(-s)γ2(x),s≤0}=-γ2(x)≤γ2(x)
which shows that assumptions (f1), (f2), and (f3) are fulfilled.



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*Source: 971243-2012-06-20.xml*