In this article we study the nonlinear Robin boundary-value problem

The aim of this article is to analyze the existence of solutions of the following problem:

We make the following assumptions on the function

where

the following limit holds uniformly for a.e

there exist two positive constants

where

By the famous Mountain Pass lemma we state the first result.

Suppose that the conditions (

Assume the following hypotheses:

We are now in the position to state our second theorem.

Suppose that

For the next theorem we assume that

Suppose that the conditions (

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 [

In [

Very recently, the authors in [

In the first result, we consider problem (

In the second result, a distinguishing feature is that we have assumed some conditions only at zero; however, there are no conditions imposed on

This article is organized as follows. First, we will introduce some basic preliminary results and lemmas in Section

For completeness, we first recall some facts on the variable exponent spaces

Both

Hölder inequality holds, namely,

Assume that the boundary of

Now, we introduce a norm, which will be used later.

Let

An important role in manipulating the generalized Lebesgue-Sobolev spaces is played by the mapping defined by the following.

Let

We recall the definition of the following condition (

Let

any bounded sequence

there exist constants

Note that condition (

Let

Here, problem (

The Euler-Lagrange functional associated with (

Standard arguments imply that

For simplicity, we use

Noting that

We check the assumption of compactness of the Mountain Pass Theorem as in the following lemma.

Lemma 10.

Let

If

From (

We will show that

Lemma

To apply the Mountain Pass Theorem, it suffices to show that

Let

Let

The main idea (developed by Wang [

Proposition

We need to state the following results.

Indeed, suppose that

In fact, let us define

On the other hand, we have

In fact, by Claim

Next, we modify and extend

For any

From Claim

Afterwards, for any

From Claims

Firstly, we recall the definition of subsupersolution of problem (

Lemma

The proof of Lemma

According to Proposition

Proposition

Let us consider the following problem:

Taking

Obviously 0 is a subsolution of (

The author declares that he has no competing interests.