Existence of Solutions for a Robin Problem Involving the p(x)-Laplace Operator

and Applied Analysis 3 Lemma 5 (see [19]). Hölder inequality holds, namely,

By the famous Mountain Pass lemma we state the first result.
Assume the following hypotheses: We are now in the position to state our second theorem.
Nonlinear boundary-value problems with variable exponent have received considerable attention in recent years.This is partly due to their frequent appearance in applications such as the modeling of electrorheological fluids [1][2][3][4] and image processing [5], but these problems are very interesting from a purely mathematical point of view as well.Many results have been obtained on this kind of problems; see for example [6][7][8][9][10][11][12][13].In [9], the authors have studied the case (, ) = || ()−2 ; they proved the existence of infinitely many eigenvalue sequences.Unlike the -Laplacian case, for a variable exponent () ( ̸ = constant), there does not exist a principal eigenvalue and the set of all eigenvalues is not closed under some assumptions.Finally, they presented some sufficient conditions that the infimum of all eigenvalues is zero and positive, respectively.
In [14], the authors obtained results on existence and multiplicity of solutions for problem (1) in the case  − >  + , under ( 0 ) and the following Ambrosetti-Rabinowitz type condition: Here, we notice that ( 3 ) is much weaker than the (AR) condition in the constant exponent case.
Very recently, the authors in [15] studied the following problem: where  is a positive parameter, (, ) is locally Lipschitz function in the -variable integrand, and (, ) is the subdifferential with respect to the -variable in the sense of Clarke.They claim that problem (6) admits at least two nontrivial solutions.
In the first result, we consider problem (1) when the nonlinear term is superlinear at infinity but does not satisfy the (AR) type condition, used in [14,16], which is necessary to ensure the boundedness of the Palais-Smale (PS) type sequences of the associated functional.To overcome these difficulties, we will use the Mountain Pass Theorem [17] with Cerami condition () which is weaker than Palais-Smale (PS) condition.
In the second result, a distinguishing feature is that we have assumed some conditions only at zero; however, there are no conditions imposed on  at infinity, which is necessary in many works.Finally, in Theorem 3, applying the subsuper solution method we get a positive solution of problem (1).
This article is organized as follows.First, we will introduce some basic preliminary results and lemmas in Section 2. In Section 3, we will give the proofs of our main results.

Preliminaries
For completeness, we first recall some facts on the variable exponent spaces  () (Ω) and  1,() (Ω).For more details, see [18,19].Suppose that Ω is a bounded open domain of R  with smooth boundary Ω and  ∈  + (Ω), where Denote by  − fl inf ∈Ω () and  + fl sup ∈Ω ().Define the variable exponent Lebesgue space  () (Ω) by with the norm Define the variable exponent Sobolev space  1,() (Ω) by with the norm We refer the reader to [11,18] for the basic properties of the variable exponent Lebesgue and Sobolev spaces.
An important role in manipulating the generalized Lebesgue-Sobolev spaces is played by the mapping defined by the following.
We recall the definition of the following condition (), see [20].
Note that condition () is weaker than the (PS) condition.However, it was shown in [17] that from condition () it is possible to obtain a deformation lemma, which is fundamental in order to get some min-max theorems.
The Euler-Lagrange functional associated with ( 1) is defined as One says that  ∈  is a weak solution of (1) if for all V ∈ .
Standard arguments imply that  ∈  1 (, R) and for all , V ∈ .Thus, the weak solutions of (1) coincide with the critical points of .

Proof of Main Results
For simplicity, we use   ,  = 1, 2, . .., to denote the general positive constants whose exact values may change from line to line.Noting that  is the sum of ( + ) type map and a weaklystrongly continuous map, so   is of ( + ) type.To see that Cerami condition () holds, it is enough to verify that any Cerami sequence is bounded.
Proof of Theorem 1.We check the assumption of compactness of the Mountain Pass Theorem as in the following lemma.
We will show that  possesses the Mountain Pass geometry.
Proof.In view of ( 0 ) and ( 2 ), there exists  1 > 0 such that Therefore, for ‖‖  ≤ 1 we have ) . ( ) is strictly positive in a neighborhood of zero.It follows that there exist  > 0 and  > 0 such that To apply the Mountain Pass Theorem, it suffices to show that  () → −∞ as  → +∞ (37) for a certain  ∈ .
It follows that there exists  ∈  such that ‖‖  >  and () < 0. According to the Mountain Pass Theorem,  admits a critical value  which is characterized by where This completes the proof.
Proof of Theorem 2. The main idea (developed by Wang [21]) is to extend  ∈ (Ω × (−, ), R) to an appropriate function f ∈ (Ω×R, R) in order to prove for the associated modified functional  the existence of a sequence of weak solutions tending to zero in  ∞ norm.Therefore, it is worth recalling the following proposition.
We need to state the following results.(44) Then we obtain which contradicts the assumption ( 5 ).
Claim 3. The associated modified functional  satisfies the Palais-Smale condition.
In fact, by Claim 2, it is easy to see that  is even and  ∈  1 (, R).For ‖‖  > 1, we have Because  →   − (Ω) with  is a positive constant,  is coercive, that is,  → +∞ as ‖‖  → +∞.Hence, to verify that  satisfies (PS) condition on , it is enough to verify that any (PS) sequence is bounded.Hence, by the coercivity of , any (PS) sequence is bounded in .
From Claim 2, for ‖‖  < 1 we can obtain By (53) and as it is well known that all norms in   are equivalent, for sufficiently small   and suitable positive constant  we obtain sup As a consequence of this fact, we observe that the conditions of Proposition 12 hold and thus there exists a sequence of negative critical values   for the functional  such that   → 0 as  → ∞.Afterwards, for any   ∈  satisfying (  ) =   and   (  ) = 0, {  }  is (PS) 0 sequence of .Passing, if necessary, to a subsequence still denoted by {  }  , we may suppose that {  }  has a limit.