By using the variational method, under appropriate assumptions on the perturbation
terms
f(x,u),g(x,u)
such that the associated functional satisfies the global minimizer
condition and the fountain theorem, respectively, the existence and multiple results for the
p(x)-Laplacian with nonlinear boundary condition in bounded domain Ω were studied. The
discussion is based on variable exponent Lebesgue and Sobolev spaces.

1. Introduction

In recent years, increasing attention has been paid to the study of differential and partial differential equations involving variable exponent conditions. The interest in studying such problems was stimulated by their applications in elastic mechanics, fluid dynamics, or calculus of variations. For more information on modeling physical phenomena by equations involving p(x)-growth condition we refer to [1–3]. The appearance of such physical models was facilitated by the development of variable exponent Lebesgue and Sobolev spaces, Lp(x) and W1,p(x), where p(x) is a real-valued function. Variable exponent Lebesgue spaces appeared for the first time in the literature as early as 1931 in an article by Orlicz [4]. The spaces Lp(x) are special cases of Orlicz spaces Lφ originated by Nakano [5] and developed by Musielak and Orlicz [6, 7], where f∈Lφ if and only if ∫φ(x,|f(x)|)dx<∞ for a suitable φ. Variable exponent Lebesgue spaces on the real line have been independently developed by Russian researchers. In that context we refer to the studies of Tsenov [8], Sharapudinov [9], and Zhikov [10, 11].

In this paper, we consider the following nonlinear elliptic boundary value problem:
(1.1)-div(a(x)|∇u|p(x)-2∇u)+b(x)|u|p(x)-2u=λf(x,u),x∈Ω,a(x)|∇u|p(x)-2∂u∂ν=c(x)|u|q(x)-2u+μg(x,u),x∈∂Ω,
where Ω⊂ℝn is a bounded domain with Lipschitz boundary ∂Ω,∂/∂ν is outer unit normal derivative, p(x)∈C(Ω¯),q(x)∈C(∂Ω),p(x),q(x)>1, and p(x)≠q(y) for any x∈Ω,y∈∂Ω;λ,μ∈ℝ;f:Ω×ℝ→ℝ, and g:∂Ω×ℝ→ℝ are Carathédory functions. Throughout this paper, we assume that a(x),b(x), and c(x) satisfy 0<a1≤a(x)≤a2,0<b1≤b(x)≤b2, and 0≤c1≤c(x)≤c2.

The operator -Δp(x)u:=-div(|∇u|p(x)-2∇u) is called p(x)-Laplacian, which is a natural extension of the p-Laplace operator, with p being a positive constant. However, such generalizations are not trivial since the p(x)-Laplace operator possesses a more complicated structure than the p-Laplace operator, for example, it is inhomogeneous. For related results involving the Laplace operator, see [12, 13].

In the past decade, many people have studied the nonlinear boundary value problems involving p-Laplacian. For example, if λ=μ=1,a(x)=b(x)=c(x)≡1,p(x)≡p, and q(x)≡q (a constant), then problem (1.1) becomes
(1.2)-div(|∇u|p-2∇u)+|u|p-2u=f(x,u),x∈Ω,|∇u|p-2∂u∂ν=|u|q-2u+g(x,u),x∈∂Ω.
Bonder and Rossi [14] considered the existence of nontrivial solutions of problem (1.2) when f(x,u)≡0 and discussed different cases when g(x,u) is subcritical, critical, and supercritical with respect to u. We also mention that Martínez and Rossi [15] studied the existence of solutions when p=q and the perturbation terms f(x,u) and g(x,u) satisfy the Landesman-Lazer-type conditions. Recently, J.-H. Zhao and P.-H. Zhao [16] studied the nonlinear boundary value problem, assumed that f(x,u) and g(x,u) satisfy the Ambrosetti-Rabinowitz-type condition, and got the multiple results.

If λ=μ=1,p(x)≡p, and q(x)≡q (a constant), then problem (1.1) becomes
(1.3)-div(a(x)|∇u|p-2∇u)+b(x)|u|p-2u=f(x,u),x∈Ω,a(x)|∇u|p-2∂u∂ν=c(x)|u|q-2u+g(x,u),x∈∂Ω.
There are also many people who studied the p-Laplacian nonlinear boundary value problems involving (1.3). For example, Cîrstea and Rǎdulescu [17] used the weighted Sobolev space to discuss the existence and nonexistence results and assumed that f(x,u) is a special case in the problem (1.3), where Ω is an unbounded domain. Pflüger [18], by using the same technique, considered the existence and multiplicity of solutions when b(x)≡0. The author showed the existence result when f(x,u) and g(x,u) are superlinear and satisfy the Ambrosetti-Rabinowitz-type condition and got the multiplicity of solutions when one of f(x,u) and g(x,u) is sublinear and the other one is superlinear.

More recently, the study on the nonlinear boundary value problems with variable exponent has received considerable attention. For example, Deng [19] studied the eigenvalue of p(x)-Laplacian Steklov problem, and discussed the properties of the eigenvalue sequence under different conditions. Fan [20] discussed the boundary trace embedding theorems for variable exponent Sobolev spaces and some applications. Yao [21] constrained the two nonlinear perturbation terms f(x,u) and g(x,u) in appropriate conditions and got a number of results for the existence and multiplicity of solutions. Motivated by Yao and problem (1.3), we consider the more general form of the variable exponent boundary value problem (1.1). Under appropriate assumptions on the perturbation terms f(x,u) and g(x,u), by using the global minimizer method and fountain theorem, respectively, the existence and multiplicity of solutions of (1.1) were obtained. These results extend some of the results in [21] and the classical results for the p-Laplacian in [14, 16, 22–24].

2. Preliminaries

In order to discuss problem (1.1), we need some results for the spaces W1,p(x)(Ω), which we call variable exponent Sobolev spaces. We state some basic properties of the spaces W1,p(x)(Ω), which will be used later (for more details, see [25, 26]). Let Ω be a bounded domain of ℝn, and denote
(2.1)C+(Ω¯)={p(x)∣p(x)∈C(Ω¯);p(x)>1,∀x∈Ω¯}.
For p(x)∈C+(Ω¯) write
(2.2)p+=maxx∈Ω¯p(x),p-=minx∈Ω¯p(x).
We can also denote C+(∂Ω) and q+,q- for any q(x)∈C(∂Ω), and define
(2.3)Lp(x)(Ω)={u∣uis a measurable real-valued function,∫Ω|u(x)|p(x)dx<∞},Lp(x)(∂Ω)={∫∂Ω|u(x)|p(x)dσ<∞u∣u:∂Ω⟶Risa measurable real-valued function,∫∂Ω|u(x)|p(x)dσ<∞},
with norms on Lp(x)(Ω) and Lp(x)(∂Ω) defined by(2.4)|u|Lp(x)(Ω)=|u|p(x)=inf{λ>0:∫Ω|u(x)λ|p(x)dx≤1},|u|Lp(x)(∂Ω)=inf{τ>0:∫∂Ω|u(x)τ|p(x)dσ≤1},
where dσ is the surface measure on ∂Ω. Then, (Lp(x)(Ω),|·|p(x)) and (Lp(x)(∂Ω),|·|Lp(x)(∂Ω)) become Banach spaces, which we call variable exponent Lebesgue spaces. Let us define the space
(2.5)W1,p(x)(Ω)={u∈Lp(x)(Ω):|∇u|∈Lp(x)(Ω)},
equipped with the norm
(2.6)‖u‖=inf{λ>0:∫Ω(|∇u(x)λ|p(x)+|u(x)λ|p(x))dx≤1}.
For u∈W1,p(x)(Ω), if we define
(2.7)‖u‖′=inf{λ>0:∫Ω(a(x)|∇u(x)λ|p(x)+b(x)|u(x)λ|p(x))dx≤1},
then, from the assumptions of a(x) and b(x), it is easy to check that ∥u∥′ is an equivalent norm on W1,p(x)(Ω). For simplicity, we denote
(2.8)Γ(u)=∫Ω(a(x)|∇u|p(x)+|u|p(x))dx.

Hence, we have (see [27])

if Γ(u)≥1, then ξ1∥u∥p-≤Γ(u)≤ξ2∥u∥p+,

if Γ(u)≤1, then ζ1∥u∥p+≤Γ(u)≤ζ2∥u∥p-,

where ξ1,ξ2 and ζ1,ζ2 are positive constants independent of u.

Denote by W01,p(x)(Ω) the closure of C0∞(Ω) in W1,p(x)(Ω).

Proposition 2.1 (see [<xref ref-type="bibr" rid="B21">21</xref>, <xref ref-type="bibr" rid="B28">28</xref>]).

(1) The space (Lp(x)(Ω), |·|p(x)) is a separable, uniformly convex Banach space, and its conjugate space is Lq(x)(Ω), where 1/q(x)+1/p(x)=1. For any u∈Lp(x)(Ω) and v∈Lq(x)(Ω), one has
(2.9)|∫Ωuvdx|≤(1p-+1q-)|u|p(x)|v|q(x).

(2) If p1,p2∈C+(Ω¯), p1(x)≤p2(x), for any x∈Ω¯, then Lp2(x)(Ω)↪Lp1(x)(Ω) and the imbedding is continuous.

(1)W1,p(x)(Ω),W01,p(x)(Ω) are separable reflexive Banach spaces.

(2) If q(x)∈C+(Ω¯) and q(x)<p*(x) for any x∈Ω¯, then the embedding from W1,p(x)(Ω) into Lq(x)(Ω) is compact and continuous, where
(2.10)p*(x)={np(x)n-p(x),ifp(x)<n,∞,ifp(x)≥n.

(3) If q(x)∈C+(∂Ω) and q(x)<p*(x) for any x∈∂Ω, then the trace imbedding from W1,p(x)(Ω) into Lq(x)(∂Ω) is compact and continuous, where
(2.11)p*(x)={(n-1)p(x)n-p(x),ifp(x)<n,∞,ifp(x)≥n.

(4) (Poincaré inequality) There is a constant C>0, such that
(2.12)|u|p(x)≤C|∇u|p(x)∀u∈W01,p(x)(Ω).

If f:Ω×ℝ→ℝ is a Carathéodory function and satisfies
(2.13)|f(x,s)|≤a(x)+b|s|p1(x)/p2(x),foranyx∈Ω¯,s∈R,
where p1(x), p2(x)∈C+(Ω¯), a(x)∈Lp2(x)(Ω), a(x)≥0, and b≥0 is a constant, then the Nemytsky operator from Lp1(x)(Ω) to Lp2(x)(Ω) defined by (Nf(u))(x)=f(x,u(x)) is a continuous and bounded operator.

In the following, let X denote the generalized Sobolev space W1,p(x)(Ω), X* denote the dual space of W1,p(x)(Ω), 〈·〉 denote the dual pair, and let → represent strong convergence, ⇀ represent weak convergence, C, Ci represent the generic positive constants.

Now we state the assumptions on perturbation terms f(x,u) and g(x,u) for problem (1.1) as follows:

f:Ω×ℝ→ℝ satisfies Carathéodory condition and there exist two constants c1≥0,c2>0 such that
(3.1)|f(x,u)|≤c1+c2|u|α(x)-1,∀(x,u)∈Ω×R,
where α(x)∈C+(Ω¯) and α(x)<p*(x) for any x∈Ω¯.

There exist M1>0,θ1>p+ such that
(3.2)0<θ1F(x,u)≤f(x,u)u,|u|≥M1,∀x∈Ω.

f(x,-u)=-f(x,u),forallx∈Ω,u∈ℝ.

g:∂Ω×ℝ→ℝ satisfies Carathéodory condition and there exist two constants c1′≥0,c2′>0 such that
(3.3)|g(x,u)|≤c1′+c2′|u|β(x)-1,∀(x,u)∈∂Ω×R,
where β(x)∈C+(∂Ω) and β(x)<p*(x) for any x∈∂Ω.

There exist M2>0,θ2>p+ such that
(3.4)0<θ2G(x,u)≤g(x,u)u,|u|≥M2,∀x∈∂Ω.

g(x,-u)=-g(x,u),forallx∈∂Ω,u∈ℝ.

The functional associated with problem (1.1) is
(3.5)φ(u)=∫Ωa(x)|∇u|p(x)+b(x)|u|p(x)p(x)dx-λ∫ΩF(x,u)dx-∫∂Ωc(x)q(x)|u|q(x)dσ-μ∫∂ΩG(x,u)dσ,
where F(x,u) and G(x,u) are denoted by
(3.6)F(x,u)=∫0uf(x,s)ds,G(x,u)=∫0ug(x,s)ds.
By Propositions 3.1 and 3.2, and assumptions (f0), (g0), it is easy to see that the functional φ∈C1(X,ℝ); moreover, φ is even if (f2) and (g3) hold. Then,
(3.7)〈φ′(u),v〉=∫Ω(a(x)|∇u|p(x)-2∇u∇v+b(x)|u|p(x)-2uv)dx-λ∫Ωf(x,u)vdx-∫∂Ωc(x)|u|q(x)-2uvdσ-μ∫∂Ωg(x,u)vdσ,
so the weak solution of (1.1) corresponds to the critical point of the functional φ.

Before giving our main results, we first give several propositions that will be used later.

Proposition 3.1 (see [<xref ref-type="bibr" rid="B32">31</xref>]).

If one denotes
(3.8)I(u)=∫Ωa(x)|∇u|p(x)+b(x)|u|p(x)p(x)dx,∀u∈X,
then I∈C1(X,ℝ) and the derivative operator of I, denoted by I′, is
(3.9)〈I′(u),v〉=∫Ω(a(x)|∇u|p(x)-2∇u∇v+b(x)|u|p(x)-2uv)dx,∀u,v∈X,
and one has:

I′:X→X* is a continuous, bounded, and strictly monotone operator,

I′ is a mapping of (S+) type, that is, if un⇀u in X and limsupn→∞〈I′(un)-I′(u),un-u〉≤0, then un→u in X,

I′:X→X* is a homeomorphism.

Proposition 3.2 (see [<xref ref-type="bibr" rid="B19">19</xref>]).

If one denotes
(3.10)J(u)=∫∂Ωc(x)q(x)|u|q(x)dσ,∀u∈X,
where q(x)∈C+(∂Ω) and q(x)<p*(x) for any x∈∂Ω, then J∈C1(X,ℝ) and the derivative operator J′ of J is
(3.11)〈J′(u),v〉=∫∂Ωc(x)|u|q(x)-2uvdσ,∀u,v∈X,
and one has that J:X→ℝ and J′:X→X* are sequentially weakly-strongly continuous, namely, un⇀u in X implies J′(un)→J′(u).

Let X be a reflexive and separable Banach space. There exist ei∈X and ej*∈X* such that
(3.12)X=span{ei:i=1,2,…}¯,X*=span{ej*:j=1,2,…}¯,〈ei,ej*〉={1,i=j,0,i≠j.
For k=1,2,…, denote
(3.13)Xk=span{ek},Yk=⨁i=1kXi,Zk=⨁i≥kXi¯.

One important aspect of applying the standard methods of variational theory is to show that the functional φ satisfies the Palais-Smale condition, which is introduced by the following definition.

Definition 3.3.

Let φ∈C1(X,ℝ) and c∈ℝ. Then, functional φ satisfies the (PS)c condition if any sequence {un}⊂X such that
(3.14)φ(un)⟶c,φ′(un)⟶0inX*,asn⟶∞
contains a subsequence converging to a critical point of φ.

In what follows we write the (PS)c condition simply as the (PS) condition if it holds for every level c∈ℝ for the Palais-Smale condition at level c.

X is a Banach space, φ∈C1(X,ℝ) is an even functional, the subspaces Xk,Yk and Zk are defined by (3.13).Suppose that, for every k∈N, there exist ρk>γk>0 such that

infu∈Zk,∥u∥=γkφ(u)→∞ as k→∞,

maxu∈Yk,∥u∥=ρkφ(u)≤0,

φ satisfies (PS)c condition for every c>0.

Then, φ has a sequence of critical values tending to +∞.

Proposition 3.5 (see [<xref ref-type="bibr" rid="B21">21</xref>]).

Suppose that hypotheses α(x)∈C+(Ω¯),α(x)<p*(x),forallx∈Ω¯, and if q(x)∈C+(∂Ω),q(x)<p*(x),forallx∈∂Ω, denote
(3.15)αk=sup{|u|Lα(x)(Ω):‖u‖=1,u∈Zk};qk=sup{|u|Lq(x)(∂Ω):‖u‖=1,u∈Zk},
then limk→∞αk=0,limk→∞qk=0.

Let us introduce the following lemma that will be useful in the proof of our main result.

Lemma 3.6.

Let λ,μ≥0,q->θ1,θ2, and assume that (f0),(f1),(g0),and(g1) are satisfied, then φ satisfies (PS) condition.

Proof.

By Propositions 2.2 and 2.3, we know that if we denote
(3.16)Φ(u)=λ∫ΩF(x,u)dx+μ∫∂ΩG(x,u)dσ,
then Φ is weakly continuous and its derivative operator, denoted by Φ′, is compact. By Propositions 3.1 and 3.2, we deduce that φ′=I′-J′-Φ′ is also of (S+) type. To verify that φ satisfies (PS) condition on X, it is enough to verify that any (PS) sequence is bounded. Suppose that {un}⊂X such that
(3.17)φ(un)⟶c,φ′(un)⟶0,inX*,asn⟶∞.
Then, for n large enough, we can find M3>0 such that
(3.18)|φ(un)|≤M3.
Since φ′(un)→0, we have 〈φ′(un),un〉→0. In particular, {〈φ′(un),un〉} is bounded. Thus, there exists M4>0 such that
(3.19)|〈φ′(un),un〉|≤M4.
We claim that the sequence {un} is bounded. If it is not true, by passing a subsequence if necessary, we may assume that ∥un∥→+∞. Without loss of generality, we assume that ∥un∥≥1 appropriately large such that ξ1∥u∥p-<ζ1∥u∥p+ for any x∈Ω. From (3.18) and (3.19) and letting θ=min{θ1,θ2}, then θ<q-, we have
(3.20)M3≥φ(un)=I(un)-J(un)-Φ(un)≥1p+Γ(un)-1q-∫∂Ωc(x)|un|q(x)dσ-Φ(un),≥1p+Γ(un)-1θ∫∂Ωc(x)|un|q(x)dσ-Φ(un),(3.21)M4≥-〈φ′(un),un〉=-Γ(un)+∫∂Ωc(x)|un|q(x)dσ+〈Φ′(un),un〉.
By virtue of assumptions (f1) and (g1) and combining (3.20) and (3.21), we have
(3.22)θM3+M4≥(θp+-1)Γ(un)-θΦ(un)+〈Φ′(un),un〉≥(θp+-1)ξ1‖un‖p-+λ∫Ω(f(x,un)un-θF(x,un))dx+μ∫∂Ω(g(x,un)un-θG(x,un))dσ≥(θp+-1)ξ1‖un‖p--C.
Note that θ=min{θ1,θ2}>p+, let n→∞ we obtian a contradiction. It follows that the sequence {un} is bounded in X. Therefore, φ satisfies (PS) condition.

Under appropriate assumptions on the perturbation terms f(x,u),g(x,u), a sequence of weak solutions with energy values tending to +∞ was obtained. The main result of the paper reads as follows.

Theorem 3.7.

Let α-,β->p+,q->θ1,θ2, and λ,μ≥0, and assumed that (f0)–(f2),(g0)–(g2) are satisfied; then φ has a sequence of critical points {±un} such that φ(±un)→∞ as n→∞.

Proof.

We will prove that φ satisfies the conditions of Proposition 3.4. Obviously, because of the assumptions of (f2) and (g2), φ is an even functional and satisfies (PS) condition (see Lemma 3.6). We will prove that if k is large enough, then there exist ρk>γk>0 such that (A2) and (A3) hold. By virtue of (f0), (g0), there exist two positive constants C1,C2 such that
(3.23)|F(x,u)|≤C1(1+|u|α(x)),(x,u)∈Ω×R;|G(x,u)|≤C2(1+|u|β(x)),(x,u)∈∂Ω×R.
Letting u∈Zk with ∥u∥>1 appropriately large such that ξ1∥u∥p-<ζ1∥u∥p+, we have
(3.24)φ(u)=I(u)-J(u)-Φ(u)≥1p+Γ(u)-c2q-∫∂Ω|u|q(x)dσ-λ∫ΩC1(1+|u|α(x))dx-μ∫∂ΩC2(1+|u|β(x))dσ≥1p+min{ξ1‖u‖p-,ζ1‖u‖p+}-c2q-max{|u|Lq(x)(∂Ω)q+,|u|Lq(x)(∂Ω)q-}-λC1max{|u|Lα(x)(Ω)α+,|u|Lα(x)(Ω)α-}-μC2max{|u|Lβ(x)(∂Ω)β+,|u|Lβ(x)(∂Ω)β-}-C3≥ξ1p+‖u‖p--C(q-,λ,μ)max{|u|Lq(x)(∂Ω)q+,|u|Lq(x)(∂Ω)q-,|u|Lα(x)(Ω)α+,|u|Lα(x)(Ω)α-,|u|Lβ(x)(∂Ω)β+,|u|Lβ(x)(∂Ω)β-}-C3.
If max{|u|Lq(x)(∂Ω)q+,|u|Lq(x)(∂Ω)q-,|u|Lα(x)(Ω)α+,|u|Lα(x)(Ω)α-,|u|Lβ(x)(∂Ω)β+,|u|Lβ(x)(∂Ω)β-}=|u|Lq(x)(∂Ω)q+, then by Proposition 3.5, we have
(3.25)φ(un)≥ξ1p+‖u‖p--C(q-,λ,μ)|u|Lq(x)(∂Ω)q+-C3≥ξ1p+‖u‖p--C(q-,λ,μ)qkq+‖u‖q+-C3.
Choose γk=(q+C(q-,λ,μ)(q_k^(q^+))/ξ_1)1/(p--q+). For u∈Zk with ∥u∥=γk, we have
(3.26)φ(u)≥ξ1(1p+-1q+)γkp--C3.
Since qk→0 as k→∞ and 1<p-≤p+<θ1,θ2<q-≤q+, we have 1/p+-1/q+>0 and γk→∞. Thus, for sufficiently large k, we have φ(u)→∞ with u∈Zk and ∥u∥=γk as k→∞. In other cases, similarly, we can deduce
(3.27)φ(u)⟶∞,sinceαk⟶0,qk=0,k⟶∞.
So (A2) holds.

By virtue of (f1) and (g1), there exist two positive constants C4,C5 such that
(3.28)F(x,u)≥C4(|u|θ1-1),∀(x,u)∈Ω×R;G(x,u)≥C5(|u|θ2-1),∀(x,u)∈∂Ω×R.
Letting u∈Yk, we have
(3.29)φ(u)≤1p-Γ(u)-c1q+∫∂Ω|u|q(x)dσ-λ∫ΩF(x,u)dx-μ∫∂ΩG(x,u)dσ≤1p-max{ξ2‖u‖p+,ζ2‖u‖p-}-c1q+min{|u|Lq(x)(∂Ω)q+,|u|Lq(x)(∂Ω)q-}-C4λ∫Ω|u|θ1dx-C5μ∫∂Ω|u|θ2dσ+C6.
If max{ξ2∥u∥p+,ζ2∥u∥p-}=ξ2∥u∥p+,min{|u|Lq(x)(∂Ω)q+,|u|Lq(x)(∂Ω)q-}=|u|Lq(x)(∂Ω)q-, then we have
(3.30)φ(u)≤ξ2p-‖u‖p+-c1q+|u|Lq(x)(∂Ω)q--C4λ∫Ω|u|θ1dx-C5μ∫∂Ω|u|θ2dσ+C6.
Since dimYk<∞, all norms are equivalent in Yk. So we get
(3.31)φ(u)≤ξ2p-‖u‖p+-c1q+C7‖u‖q--C8λ‖u‖θ1-C9μ‖u‖θ2+C6.
Also, note that q->θ1,θ2>p+, Then, we get φ(u)→-∞ as ∥u∥→∞. For other cases, the proofs are similar and we omit them here. So (A3) holds. From the proof of (A2) and (A3), we can choose ρk>γk>0. Thus, we complete the proof.

This time our idea is to show that φ possesses a nontrivial global minimum point in X.

Theorem 3.8.

Let α+,β+,q+<p-, and assume (f0), (g0) are satisfied; then (1.1) has a weak solution.

Proof.

Firstly, we show that φ is coercive. For sufficiently large norm of u(∥u∥≥1), and by virtue of (3.23),
(3.32)φ(u)=∫Ωa(x)|∇u|p(x)+b(x)|u|p(x)p(x)dx-λ∫ΩF(x,u)dx-∫∂Ωc(x)q(x)|u|q(x)dσ-μ∫∂ΩG(x,u)dσ≥ξ1p+‖u‖p--|λ|∫ΩC1(1+|u|α(x))dx-c2q-∫∂Ω|u|q(x)dσ-|μ|∫∂ΩC2(1+|u|β(x))dσ≥ξ1p+‖u‖p--|λ|C1max{|u|Lα(x)(Ω)α+,|u|Lα(x)(Ω)α-}-c2q-max{|u|Lq(x)(∂Ω)q+,|u|Lq(x)(∂Ω)q-}-|μ|C2max{|u|Lβ(x)(∂Ω)β+,|u|Lβ(x)(∂Ω)β-}-C10.
If
(3.33)max{|u|Lα(x)(Ω)α+,|u|Lα(x)(Ω)α-}=|u|Lα(x)(Ω)α+,max{|u|Lq(x)(∂Ω)q+,|u|Lq(x)(∂Ω)q-}=|u|Lq(x)(∂Ω)q+,max{|u|Lβ(x)(∂Ω)β+,|u|Lβ(x)(∂Ω)β-}=|u|Lβ(x)(∂Ω)β+,
then
(3.34)φ(u)≥ξ1p+‖u‖p--C11|λ|‖u‖α+-C12‖u‖q+-C13|μ|‖u‖β+-C10⟶∞as∥u∥⟶∞.
So φ is coercive since α+,β+,q+<p-. Secondly, by Proposition 2.2, it is easy to verify that φ is weakly lower semicontinuous. Thus, φ is bounded below and φ attains its infimum in X, that is, φ(u0)=infu∈Xφ(u) and u0 is a critical point of φ, which is a weak solution of (1.1).

In the Theorem 3.8, we cannot guarantee that u0 is nontrivial. In fact, under the assumptions on the above theorem, we can also get a nontrivial weak solution of φ.

Corollary 3.9.

Under the assumptions in Theorem 3.8, if one of the following conditions holds, (1.1) has a nontrivial weak solution.

If λ,μ≠0, there exist two positive constants d1,d2<p- such that
(3.35)liminfu→0sgn(λ)F(x,u)|u|d1>0,forx∈Ωuniformly,liminfu→0sgn(μ)G(x,u)|u|d2>0,forx∈∂Ωuniformly.

If λ=0,μ≠0, there exist two positive constants d2<p- such that
(3.36)liminfu→0sgn(μ)G(x,u)|u|d2>0,forx∈∂Ωuniformly.

If λ≠0,μ=0, there exist two positive constants d1<p- such that
(3.37)limu→0infsgn(λ)F(x,u)|u|d1>0,forx∈Ωuniformly.

Proof.

From Theorem 3.8, we know that φ has a global minimum point u0. We just need to show that u0 is nontrivial. We only consider the case λ,μ≠0 here. From (1), we know that for 0<u<1 small enough, there exists two positive constants C14,C15>0 such that
(3.38)sgn(λ)F(x,u)≥C14|u|d1,sgn(μ)G(x,u)≥C15|u|d2.
Choose u-≡M>0; then u-∈X. For 0<t<1 small enough, we have
(3.39)φ(tu-)≤b2tp-p-∫Ω|u-|p(x)dx-|λ|∫Ωsgn(λ)F(x,tu-)dx-c1q+∫∂Ω|tu-|q(x)dσ-|μ|∫∂Ωsgn(μ)G(x,tu-)dσ≤b2tp-p-∫Ω|M|p(x)dx-C14|λ|td1∫Ω|M|d1dx-c1q+tq-∫∂Ω|M|q(x)dσ-C15|μ|td2∫∂Ω|M|d2dσ≤C16tp--C17|λ|td1-C18c1tq--C19|μ|td2.
Since d1,d2,<p- and q-≤q+<p-, there exists 0<t0<1 small enough such that φ(t0u-)<0. So the global minimum point u0 of φ is nontrivial.

Remark 3.10.

Suppose that f(x,u)=sgn(λ)|u|α(x)-2u,g(x,u)=sgn(μ)|u|β(x)-2u and p->α+,β+,q+; then the conditions in Corollary 3.9 can be fulfilled.

AcerbiE.MingioneG.Regularity results for a class of functionals with non-standard growthRůžičkaM.WinslowW. M.Induced fibration of suspensionsOrliczW.Über konjugierte exponentenfolgenNakanoH.MusielakJ.MusielakJ.OrliczW.On modular spacesTsenovI.Generalization of the problem of best approximation of a function in the space LsSharapudinovI. I.Topology of the space Lp(t)([0;1])ZhikovV.Averaging of functionals in the calculus of variations and elasticityZhikovV. V.Passage to the limit in nonlinear variational problemsAcerbiE.FuscoN.Partial regularity under anisotropic (p,q) growth conditionsStruweM.Three nontrivial solutions of anticoercive boundary value problems for the pseudo-Laplace operatorBonderJ. F.RossiJ. D.Existence results for the p-Laplacian with nonlinear boundary conditionsMartínezS.RossiJ. D.Weak solutions for the p-Laplacian with a nonlinear boundary condition at resonanceZhaoJ.-H.ZhaoP.-H.Infinitely many weak solutions for a p-Laplacian equation with nonlinear boundary conditionsŞt. CîrsteaF.-C.RǎdulescuV. D.Existence and non-existence results for a quasilinear problem with nonlinear boundary conditionPflügerK.Existence and multiplicity of solutions to a p-Laplacian equation with nonlinear boundary conditionDengS.-G.Eigenvalues of the p(x)-Laplacian Steklov problemFanX.Boundary trace embedding theorems for variable exponent Sobolev spacesYaoJ.Solutions for Neumann boundary value problems involving p(x)-Laplace operatorsBonderJ. F.Multiple solutions for the p-Laplace equation with nonlinear boundary conditionsWillemM.SongS.-Z.TangC.-L.Resonance problems for the p-Laplacian with a nonlinear boundary conditionFanX.Solutions for p(x)-Laplacian Dirichlet problems with singular coefficientsFanX.ZhaoD.On the spaces Lp(x)(Ω) and Wm,p(x)(Ω)LiuW.ZhaoP.Existence of positive solutions for p(x)-Laplacian equations in unbounded domainsFanX. L.ZhaoD.On the generalized Olicz-Sobolev space Wk,p(x)(Ω)ZhaoD.QiangW. J.FanX. L.On the generalized Orlicz spaces Lp(x)(Ω)ZhaoD.FanX. L.The Nemytskiĭ operators from Lp1(x)(Ω) to Lp2(x)(Ω)FanX.p(x)-Laplacian equations in ℝN with periodic data and nonperiodic perturbationsFanX.HanX.Existence and multiplicity of solutions for p(x)-Laplacian equations in ℝN