AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 10.1155/2016/2349172 2349172 Research Article Existence of Solutions for a Robin Problem Involving the p ( x ) -Laplace Operator Allaoui Mostafa 1 Wong Patricia J. Y. Department of Mathematics Faculty of sciences Department of Mathematics Mohamed I University 60000 Oujda Morocco 2016 1362016 2016 27 01 2016 09 05 2016 2016 Copyright © 2016 Mostafa Allaoui. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this article we study the nonlinear Robin boundary-value problem - Δ p ( x ) u = f ( x , u ) i n Ω , | u | p x - 2 ( u / ν ) + β ( x ) u p ( x ) - 2 u = 0 on Ω . Using the variational method, under appropriate assumptions on f , we obtain results on existence and multiplicity of solutions.

1. Introduction

The aim of this article is to analyze the existence of solutions of the following problem: (1) - Δ p x u = f x , u i n Ω , u p x - 2 u ν + β x u p x - 2 u = 0 o n Ω , where Ω R N ( N 2 ) is a bounded smooth domain, u / ν is the outer unit normal derivative on Ω , p is a continuous function on Ω ¯ with p - inf x Ω ¯ p ( x ) > 1 , and β L ( Ω ) with β - i n f x Ω β ( x ) > 0 and f : Ω × R R is a continuous function. The main interest in studying such problems arises from the presence of the p ( x ) -Laplace operator div ( u p ( x ) - 2 u ) , which is a natural extension of the classical p -Laplace operator div ( u p - 2 u ) obtained in the case when p is a positive constant. However, such generalizations are not trivial since the p ( x ) -Laplace operator possesses a more complicated structure than p -Laplace operator; for example, it is inhomogeneous.

We make the following assumptions on the function f :

f : Ω × R R is a continuous function and there exist two constants C 1 , C 2 0 such that (2) f x , s C 1 + C 2 s q x - 1 x , s Ω × R ,

where q ( x ) C ( Ω ) and 1 < q ( x ) < p ( x ) for all x Ω .

the following limit holds uniformly for a.e x Ω : (3) l i m s f x , s s s p + = + ;

f ( x , s ) = o ( s p ( x ) - 1 ) as s 0 and uniformly for x Ω .

there exist two positive constants c 1 and c 2 such that (4) ζ 1 x , s c 1 ζ 1 x , t c 2 ζ 2 x , t , 0 s t ,

where ζ 1 ( x , s ) = f ( x , s ) s - p - F ( x , s ) and ζ 2 ( x , s ) = f ( x , s ) s - p + F ( x , s ) .

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

Theorem 1.

Suppose that the conditions ( H 0 )–( H 3 ) with q - > p + hold. Then problem (1) has at least a nontrivial weak solution.

Assume the following hypotheses:

(5) lim s 0 F x , s s p - = ;

f ( x , 0 ) = 0 , and there exists δ > 0 such that p - F ( x , s ) - f ( x , s ) s > 0 for every x Ω ¯ and s δ where F ( x , s ) = 0 s f ( x , t ) d t .

f ( x , - s ) = - f ( x , s ) , for x Ω , s R .

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

Theorem 2.

Suppose that p : Ω ¯ R is Lipschitz continuous function. Under the assumptions ( H 0 ) and ( H 4 )–( H 6 ), problem (1) has a sequence of weak solutions u n n such that u n L ( Ω ) 0 as n .

For the next theorem we assume that f satisfies the following conditions:

f ( x , s ) 0 for all x Ω and s 0 with f ( x , 0 ) 0 .

f ( x , s ) is nondecreasing with respect to s 0 , x Ω .

Theorem 3.

Suppose that the conditions ( H 0 ), ( H 7 ), and ( H 8 ) with p C 1 ( Ω ¯ ) hold. Then problem (1) has a positive solution.

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  and image processing , 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 . In , the authors have studied the case f ( x , u ) = u p ( x ) - 2 u ; they proved the existence of infinitely many eigenvalue sequences. Unlike the p -Laplacian case, for a variable exponent p ( x ) ( 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 , the authors obtained results on existence and multiplicity of solutions for problem (1) in the case q - > p + , under ( H 0 ) and the following Ambrosetti-Rabinowitz type condition: A R t M 0 μ F x , t f x , t t , μ > p + , M > 0   such that   x Ω . Here, we notice that ( H 3 ) is much weaker than the A R condition in the constant exponent case.

Very recently, the authors in  studied the following problem: (6) - Δ p x u λ F x , u i n Ω , u p x - 2 u ν + β x u p x - 2 u = 0 o n Ω , where λ is a positive parameter, F ( x , t ) is locally Lipschitz function in the t -variable integrand, and F ( x , t ) is the subdifferential with respect to the t -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 A R 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  with Cerami condition ( C ) 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 f 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.

2. Preliminaries

For completeness, we first recall some facts on the variable exponent spaces L p ( x ) ( Ω ) and W 1 , p ( x ) ( Ω ) . For more details, see [18, 19]. Suppose that Ω is a bounded open domain of R N with smooth boundary Ω and p C + ( Ω ¯ ) , where (7) C + Ω ¯ = p C Ω ¯ ,   inf x Ω ¯ p x > 1 . Denote by p - i n f x Ω ¯ p ( x ) and p + s u p x Ω ¯ p ( x ) . Define the variable exponent Lebesgue space L p ( x ) ( Ω ) by (8) L p x Ω = u : Ω R   is  measurable  and   Ω u p x d x < + , with the norm (9) u p x = inf τ > 0 ; Ω u τ p x d x 1 . Define the variable exponent Sobolev space W 1 , p ( x ) ( Ω ) by (10) W 1 , p x Ω = u L p x Ω : u L p x Ω , with the norm (11) u = inf τ > 0 ; Ω u τ p x + u τ p x d x 1 , u = u p x + u p x . We refer the reader to [11, 18] for the basic properties of the variable exponent Lebesgue and Sobolev spaces.

Lemma 4 (see [<xref ref-type="bibr" rid="B13">19</xref>]).

Both ( L p x Ω , · p x ) and ( W 1 , p ( x ) ( Ω ) , · ) are separable and uniformly convex Banach spaces.

Lemma 5 (see [<xref ref-type="bibr" rid="B13">19</xref>]).

Hölder inequality holds, namely, (12) Ω u v d x 2 u p x v p x u L p x Ω , v L p x Ω , where 1 / p x + 1 / p x = 1 .

Lemma 6 (see [<xref ref-type="bibr" rid="B12">18</xref>]).

Assume that the boundary of Ω possesses the cone property and p C ( Ω ¯ ) and 1 q ( x ) < p ( x ) for x Ω ¯ , then there is a compact embedding W 1 , p ( x ) ( Ω ) L q ( x ) ( Ω ) , where (13) p x = N p x N - p x , i f p x < N ; + , i f p x N .

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

Let β L ( Ω ) with β - i n f x Ω β ( x ) > 0 and, for u W 1 , p ( x ) ( Ω ) , define (14) u β = inf τ > 0 ; Ω u τ p x d x + Ω β x u τ p x d σ 1 . Then, by Theorem   2.1 in , · β is also a norm on W 1 , p ( x ) ( Ω ) which is equivalent to · .

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

Lemma 7 (see [<xref ref-type="bibr" rid="B10">16</xref>]).

Let I ( u ) = Ω | u | p ( x ) d x + Ω β ( x ) u p ( x ) d σ with β - > 0 . For u W 1 , p ( x ) ( Ω ) one has

u β < 1 ( = 1 , > 1 ) I ( u ) < 1 ( = 1 , > 1 ) ;

u β 1 u β p + I ( u ) u β p - ;

u β 1 u β p - I ( u ) u β p + ;

u n - u β 0 I ( u n - u ) 0 .

We recall the definition of the following condition ( C ), see .

Definition 8 (see [<xref ref-type="bibr" rid="B5">20</xref>]).

Let X be a Banach space and J C 1 ( X , R ) . Given c R , one says that J satisfies the Cerami c condition (one denotes condition ( C c )) if

any bounded sequence u n X such that J ( u n ) c and J ( u n ) 0 has a convergent subsequence;

there exist constants δ , R , β > 0 such that (15) J u u β u J - 1 c - δ , c + δ   with   u R .

If J C 1 ( X , R ) satisfies condition ( C c ) for every c R , one says that J satisfies condition ( C ).

Note that condition ( C ) is weaker than the (PS) condition. However, it was shown in  that from condition ( C ) it is possible to obtain a deformation lemma, which is fundamental in order to get some min-max theorems.

Theorem 9 (see [<xref ref-type="bibr" rid="B4">17</xref>]).

Let X a Banach space, J C 1 ( X , R ) , e X , and r > 0 , such that e > r and (16) b inf u = r J u > J 0 J e . If J satisfies the condition ( C c ) with (17) c inf γ Γ max t 0,1 J γ t , Γ γ C 0,1 , X γ 0 = 0 , γ 1 = e . Then c is a critical value of J .

Here, problem (1) is stated in the framework of the generalized Sobolev space X W 1 , p ( x ) ( Ω ) .

The Euler-Lagrange functional associated with (1) is defined as J : X R in (18) J u = Ω 1 p x u p x d x + Ω β x p x u p x d σ - Ω F x , u d x . One says that u X is a weak solution of (1) if (19) Ω u p x - 2 u v d x + Ω β x u p x - 2 u v d σ - Ω f x , u v d x = 0 , for all v X .

Standard arguments imply that J C 1 ( X , R ) and (20) J u , v = Ω u p x - 2 u v d x + Ω β x u p x - 2 u v d σ - Ω f x , u v d x , for all u , v X . Thus, the weak solutions of (1) coincide with the critical points of J .

3. Proof of Main Results

For simplicity, we use C i , i = 1,2 , , to denote the general positive constants whose exact values may change from line to line.

Noting that J is the sum of ( S + ) type map and a weakly-strongly continuous map, so J is of ( S + ) type. To see that Cerami condition ( C ) holds, it is enough to verify that any Cerami sequence is bounded.

Proof of Theorem <xref ref-type="statement" rid="thm1.1">1</xref>.

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

Lemma  10. Suppose that ( H 0 )–( H 3 ) hold. If c R , then any ( C ) c sequence of J is bounded.

Proof. Let u n be a ( C ) c sequence of J . If u n is unbounded, up to a subsequence we may assume that (21) J u n c , u n β , J u n u n β 0 .

Let w n = u n β - 1 u n , then w n is bounded in X ; up to a subsequence we have (22) w n w in   X , w n w in   L q x Ω , w n x w x a.e   x Ω . If w 0 , we have J ( u n ) u n = 0 ; that is, (23) Ω u n p x d x + Ω β x u n p x d σ - Ω f x , u n u n d x = 0 . Dividing (23) by u n β p + , we get (24) Ω f x , u n u n u n β p + d x < . On the other side, using ( H 1 ) and lemma of Fatou we obtain (25) Ω f x , u n u n u n β p + d x = Ω f x , u n u n w n p + u n p + d x ; we obtain a contradiction.

If w 0 , since w n 0 in L q ( x ) ( Ω ) and F x , t C ( 1 + t q ( x ) ) , by the continuity of the Nemitskii operator, we see that F ( · , w n ) 0 in L 1 ( Ω ) as n + ; therefore, (26) lim n Ω F x , w n d x = 0 . We choose a sequence t n [ 0,1 ] such that (27) J t n u n = max t 0,1 J t u n . Given m > 0 , since for n large enough we have u n β - 1 ( 2 m p + ) 1 / p - ( 0,1 ) , using (26) with R = ( 2 m p + ) 1 / p - , we obtain (28) J t n u n J R u n β u n = J R w n = Ω R p x p x w n p x d x + Ω R p x p x β x w n p x d σ - Ω F x , R w n d x R p - p + - Ω F x , R w n d x m . That is, J ( t n u n ) + , but J ( 0 ) = 0 , J ( u n ) c ; we see that t n ( 0,1 ) and J ( t n u n ) , t n u n = t n ( d / d t ) t = t n J ( t u n ) = 0 . It yields (29) J t n u n - 1 p - J t n u n · t n u n + . Therefore, (30) Ω 1 p x - 1 p - t n u n p x d x + Ω 1 p x - 1 p - β x t n u n p x d σ + Ω 1 p - f x , t n u n t n u n - F x , t n u n d x + , so we get (31) Ω 1 p - f x , t n u n t n u n - F x , t n u n d x + . Appropriately, we have (32) J u n = J u n - 1 p + J u n · u n = Ω 1 p x - 1 p + u n p x d x + Ω 1 p x - 1 p + β x u n p x d σ + Ω 1 p + f x , u n u n - F x , u n d x Ω 1 p + f x , u n u n - F x , u n d x .

From ( H 3 ), there exist two constants c 1 and c 2 such that (33) J u n Ω 1 p + f x , u n u n - F x , u n d x c 1 Ω 1 p - f x , u n u n - F x , u n d x c 1 c 2 Ω 1 p - f x , t n u n t n u n - F x , t n u n d x . Hence, J ( u n ) + , which is impossible and thus ( u n ) n is bounded in X .

We will show that J possesses the Mountain Pass geometry.

Lemma 11 . Under the conditions ( H 0 )–( H 2 ), there exist r > 0 and τ such that J ( u ) > τ when u β = r .

Proof. In view of ( H 0 ) and ( H 2 ), there exists C 1 > 0 such that (34) F x , t 1 2 p + t p x + C 1 t q x , for x , t Ω × R . Therefore, for u β 1 we have (35) J u 1 p + Ω u p x d x + 1 p + Ω β x u p x d σ - 1 2 p + Ω u p x d x - C 1 Ω u q x d x C 2 2 p + Ω u p x d x + C 2 2 p + Ω β x u p x d σ - C 1 Ω u q x d x C 2 2 p + u β p + - C 3 u β q - u β p + C 2 2 p + - C 3 u β q - - p + . Since p + < q - , the function t ( C 2 / 2 p + - C 3 t q - - p + ) is strictly positive in a neighborhood of zero. It follows that there exist r > 0 and τ > 0 such that (36) J u τ u X : u = r .

To apply the Mountain Pass Theorem, it suffices to show that (37) J t u - as   t + for a certain u X .

Let u X 0 ; by ( H 1 ), we can choose a constant A > Ω ( 1 / p x ) u p x d x + Ω ( β ( x ) / p ( x ) ) | u | p x d σ / Ω | u | p + d x , such that (38) F x , t A t p + uniformly in   x Ω .

Let t > 1 be large enough; we have (39) J t u = Ω t p x p x u p x d x + Ω t p x β x p x u p x d σ - Ω F x , t u d x t p + Ω 1 p x u p x d x + Ω β x p x u p x d σ - t u > C A F x , t u d x - t u C A F x , t u d x t p + Ω 1 p x u p x d x + Ω β x p x u p x d σ - A t p + Ω u p + d x - t u C A F x , t u d x + A t p + t u C A u p + d x t p + Ω 1 p x u p x d x + Ω β x p x u p x d σ - A t p + Ω u p + d x + C 1 , where C 1 > 0 is a constant, which implies that (40) J t u - as t + . It follows that there exists e X such that e β > r and J ( e ) < 0 . According to the Mountain Pass Theorem, J admits a critical value c which is characterized by (41) c = i n f γ Γ sup t 0,1 J γ t , where (42) Γ = γ C 0,1 , X : γ 0 = 0 ,    γ 1 = e . This completes the proof.

Proof of Theorem <xref ref-type="statement" rid="thm1.2">2</xref>.

The main idea (developed by Wang ) is to extend f C ( Ω × ( - ϵ , ϵ ) , R ) to an appropriate function f ~ C ( Ω × R , R ) in order to prove for the associated modified functional J ¯ the existence of a sequence of weak solutions tending to zero in L norm. Therefore, it is worth recalling the following proposition.

Proposition 12 (see ). Let J C 1 ( X , R ) , where X is a Banach space. Assume that J satisfies the ( P S ) condition and is even and bounded from below, and J ( 0 ) = 0 . If for any n N , there exists a k -dimensional subspace X n and ρ n > 0 such that (43) sup X n S ρ k J < 0 , where S ρ : = u X : u = ρ , then J has a sequence of critical values c n < 0 satisfying c n 0 as n .

We need to state the following results.

Claim 1. When J u = J u · u = 0 , then u = 0 .

Indeed, suppose that u 0 . Thus (44) Ω F x , u d x = Ω 1 p x u p x d x + Ω β x p x u p x d σ , Ω u p x d x + Ω β x u p x d σ = Ω f x , u u d x . Then we obtain (45) p - Ω F x , u d x = p - Ω 1 p x u p x d x + Ω β x p x u p x d σ Ω f x , u u d x , which contradicts the assumption ( H 5 ).

Claim 2. There exist δ > 0 and f ~ C ( Ω × R ) such that f ~ is odd and (46) f x , t = f ~ x , t for   t < δ , (47) p - F ~ x , t - f ~ x , t t 0 , x , t Ω × R , (48) p - F ~ x , t - f ~ x , t t = 0 , for   t > δ or t = 0 , where F ~ ( x , t ) = 0 t f ~ ( x , s ) d s .

In fact, let us define F ~ ( x , t ) = h ( t ) F ( x , t ) + C 1 ( 1 - h ( t ) ) t p - where C 1 is a positive constant and h is a cut-off function presented as follows: (49) h t = 1 , if   t δ 2 ; 0 , if   t δ , h t t 0 , h t 4 δ . For | t | δ / 2 , (46) easily holds.

On the other hand, we have (50) f ~ x , t = t F ~ x , t = h t F x , t + h t f x , t + C 1 1 - h t t p - - C 1 h t t p - . It is easy to check that for | t | δ we have (51) p - F ~ x , t = C 1 p - t p - . Hence, (48) is satisfied. In the rest, from ( H 4 ), we can choose δ > 0 small enough to get F ( x , t ) C 1 t p - when t [ δ / 2 , δ ] and the formula (47) holds since h ( t ) t 0 .

Claim 3. The associated modified functional J ¯ satisfies the Palais-Smale condition.

In fact, by Claim 2 , it is easy to see that J ¯ is even and J ¯ C 1 ( X , R ) . For u β > 1 , we have (52) J ¯ = Ω 1 p x u p x d x + Ω β x p x u p x d σ - Ω F ~ x , u d x 1 p + Ω u p x d x + Ω β x u p x d σ - A Ω u p - d x . Because X L p - ( Ω ) with A is a positive constant, J ¯ is coercive, that is, J ¯ + as u β + . Hence, to verify that J ¯ satisfies ( P S ) condition on X , it is enough to verify that any ( P S ) sequence is bounded. Hence, by the coercivity of J ¯ , any ( P S ) sequence is bounded in X .

Next, we modify and extend f ( x , u ) to get f ~ ( x , u ) C ( Ω ¯ × R ) satisfying the assertions of Proposition 12 .

For any k N we have k independent smooth functions e i for i = 1,2 , , k , and define the subspace X k : = span e 1 , , e k .

From Claim 2 , for u β < 1 we can obtain (53) J ¯ 1 p - Ω u p x d x + Ω β x u p x d σ - C Ω u p - d x 1 p - u β p - - C Ω u p - d x . By (53) and as it is well known that all norms in X k are equivalent, for sufficiently small ρ k and suitable positive constant C we obtain (54) sup X k S ρ n J ¯ < 0 . 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 c k for the functional J ¯ such that c k 0 as k .

Afterwards, for any u k X satisfying J ¯ ( u k ) = c k and J ¯ ( u k ) = 0 , u k k is ( P S ) 0 sequence of J ¯ . Passing, if necessary, to a subsequence still denoted by u k k , we may suppose that u k k has a limit.

From Claims 1 and 2 it is clear that 0 is the only critical point when the energy is zero and thus u k k converges to 0. It follows from [23, 24] that (55) u k k C Ω ¯ , u k L Ω 0 as   k . So in view of Claim 2 , we have | u k | C ( Ω ) δ / 2 . Thereby, the sequences u k k are solutions of problem (1).

Proof of Theorem <xref ref-type="statement" rid="thm1.3">3</xref>.

Firstly, we recall the definition of subsupersolution of problem (1) as follows. We call u X a subsolution (resp. supersolution) of (1) if, for every v X with v 0 , (56) Ω u p x - 2 u v d x + Ω β x u p x - 2 u v d σ resp . Ω f x , u v d x .

Lemma 13 . Let p C 1 ( Ω ¯ ) . Suppose that f satisfies the subcritical growth condition ( H 0 ) and the function f ( x , t ) is nondecreasing in t R . If there exist a subsolution u 0 W 1 , p ( x ) ( Ω ) L ( Ω ) and a supersolution v 0 W 1 , p ( x ) ( Ω ) L ( Ω ) of (1) such that u 0 v 0 , then (1) has a minimal solution u _ and a maximal solution v ¯ in the order interval [ u 0 , v 0 ] (i.e., u 0 u _ v ¯ v 0 ).

The proof of Lemma 13 is built on the fixed point theory for the increasing operator on the order interval (see e.g., ) and is similar to that given in  for the p ( x ) -Laplacian case.

According to Proposition 2.2 in , the mapping I : X X such that for all u , v in X ; I ( u ) , v = Ω u p ( x ) - 2 u v d x + Ω β ( x ) | u | p ( x ) - 2 u v d σ is a strictly monotone, bounded homeomorphism, and consequently we have the following.

Proposition 14 . Let q C ( Ω ¯ ) with 1 < q ( x ) < p ( x ) for x Ω ¯ , then for g L ( q x ) / ( q x - 1 ) ( Ω ) (or g C 0 , α ( Ω ) ), the problem P 1 - Δ p x u = g x , i n Ω ; u p x - 2 u ν + β x u p x - 2 u = 0 , o n    Ω , has a unique solution u in X .

Let us consider the following problem: P 2 - Δ p x u = M , in   Ω ; u p x - 2 u ν + β x u p x - 2 u = 0 , on   Ω , with M > 0 . By Proposition 14 , the strong maximum principle  and the result of regularity in , problem P 2 has a unique positive solution u 1 such that u 1 ( x ) > 0 for each x Ω ¯ .

Taking M = s u p x Ω ¯ f ( x , t ) , for any v X with v 0 we have (57) Ω u 1 p x - 2 u 1 v d x + Ω β x u 1 p x - 2 u 1 v d σ = Ω M v d x Ω f x , u 1 v d x . Hence, u 1 is a positive supersolution of problem (1).

Obviously 0 is a subsolution of (1). By Lemma 13 , (1) has a solution u [ 0 , u 1 ] .

Competing Interests

The author declares that he has no competing interests.

Myers T. G. Thin films with high surface tension SIAM Review 1998 40 3 441 462 10.1137/S003614459529284X ZBL0908.35057 2-s2.0-0032165809 Rŭžicka M. Electrorheological Fluids: Modeling and Mathematical Theory 2000 Berlin, Germany Springer Zhikov V. V. Averaging of functionals of the calculus of variations and elasticity theory Mathematics of the USSR-Izvestiya 1987 29 1 33 66 10.1070/im1987v029n01abeh000958 Zhikov V. V. Kozlov S. M. Oleinik O. A. Homogenization of Differential Operators and Integral Functionals Berlin, Germany Springer Translated from Russian by G. A. Yosifian, 1994 Chen Y. M. Levine S. Rao M. Variable exponent, linear growth functionals in image restoration SIAM Journal on Applied Mathematics 2006 66 4 1383 1406 10.1897/05-356r.1 2-s2.0-33747163537 Allaoui M. Existence of solutions for a Robin problem involving the p x -Laplacian Applied Mathematics E—Notes 2014 14 107 115 2-s2.0-84908302603 Allaoui M. Continuous spectrum of steklov nonhomogeneous elliptic problem Opuscula Mathematica 2015 35 6 853 866 10.7494/OpMath.2015.35.6.853 ZBL1333.35051 2-s2.0-84930520022 Chabrowski J. Fu Y. Existence of solutions for p(x)-Laplacian problems on a bounded domain Journal of Mathematical Analysis and Applications 2005 306 2 604 618 10.1016/j.jmaa.2004.10.028 Deng S.-G. Wang Q. Cheng S. On the p x -Laplacian Robin eigenvalue problem Applied Mathematics and Computation 2011 217 12 5643 5649 10.1016/j.amc.2010.12.042 2-s2.0-79551617513 Deng S.-G. A local mountain pass theorem and applications to a double perturbed p(x)-Laplacian equations Applied Mathematics and Computation 2009 211 1 234 241 10.1016/j.amc.2009.01.042 2-s2.0-64049108465 Shi X. Ding X. Existence and multiplicity of solutions for a general p x -Laplacian Neumann problem Nonlinear Analysis: Theory, Methods & Applications 2009 70 10 3715 3720 10.1016/j.na.2008.07.027 2-s2.0-61749100674 Ourraoui A. Multiplicity results for Steklov problem with variable exponent Applied Mathematics and Computation 2016 277 34 43 10.1016/j.amc.2015.12.043 Wang L.-L. Fan Y.-H. Ge W.-G. Existence and multiplicity of solutions for a Neumann problem involving the p ( x ) -Laplace operator Nonlinear Analysis: Theory, Methods and Applications 2009 71 9 4259 4270 10.1016/j.na.2009.02.116 2-s2.0-67349203616 Tsouli N. Darhouche O. Existence and multiplicity results for nonlinear problems involving the p(x)-Laplace operator Opuscula Mathematica 2014 34 3 621 638 10.7494/opmath.2014.34.3.621 2-s2.0-84904661435 Ge B. Zhou Q.-M. Multiple solutions for a Robin-type differential inclusion problem involving the p(x)-Laplacian Mathematical Methods in the Applied Sciences 2013 10.1002/mma.2760 2-s2.0-84874247965 Deng S.-G. Positive solutions for Robin problem involving the p ( x ) -Laplacian Journal of Mathematical Analysis and Applications 2009 360 2 548 560 10.1016/j.jmaa.2009.06.032 2-s2.0-70349382551 Bartolo P. Benci V. Fortunato D. Abstract critical point theorems and applications to some nonlinear problems with ‘strong’ resonance at infinity Nonlinear Analysis 1983 7 9 981 1012 10.1016/0362-546x(83)90115-3 2-s2.0-0009179578 Fan X. L. Shen J. S. Zhao D. Sobolev embedding theorems for spaces W k , p x Ω Journal of Mathematical Analysis and Applications 2001 262 2 749 760 10.1006/jmaa.2001.7618 2-s2.0-0035888627 Fan X. L. Zhao D. On the spaces L p x (Ω) and W m , p x (Ω) Journal of Mathematical Analysis and Applications 2001 263 2 424 446 10.1006/jmaa.2000.7617 Cerami G. An existence criterion for the critical points on unbounded manifolds Istituto Lombardo Accademia di Scienze e Lettere Rendiconti A 1978 112 2 332 336 Wang Z.-Q. Nonlinear boundary value problems with concave nonlinearities near the origin Nonlinear Differential Equations and Applications 2001 8 1 15 33 10.1007/pl00001436 2-s2.0-0002823383 Heinz H.-P. Free Ljusternik-Schnirelman theory and the bifurcation diagrams of certain singular nonlinear problems Journal of Differential Equations 1987 66 2 263 300 10.1016/0022-0396(87)90035-0 2-s2.0-0001327974 Fan X. L. Zhao D. A class of De Giorgi type and Hölder continuity Nonlinear Analysis: Theory, Methods & Applications 1999 36 3 295 318 10.1016/s0362-546x(97)00628-7 2-s2.0-0033132638 Fan X. Global C1, α regularity for variable exponent elliptic equations in divergence form Journal of Differential Equations 2007 235 2 397 417 10.1016/j.jde.2007.01.008 2-s2.0-33847727326 Amann H. Fixed point equations and nonlinear eigenvalue problems in ordered banach spaces SIAM Review 1976 18 4 620 709 10.1137/1018114 2-s2.0-0017014016 Fan X. L. On the sub-supersolution method for p ( x ) -Laplacian equations Journal of Mathematical Analysis and Applications 2007 330 1 665 682 10.1016/j.jmaa.2006.07.093 2-s2.0-33846903872 Fan X. L. Zhao Y. Z. Zhang Q. H. A strong maximum principle for p(x)-Laplace equations Chinese Journal of Contemporary Mathematics 2003 24 277 282 Fan X. Global C 1 ,   α regularity for variable exponent elliptic equations in divergence form Journal of Differential Equations 2007 235 2 397 417 10.1016/j.jde.2007.01.008 2-s2.0-33847727326