PERTURBATION EFFECTS FOR THE MINIMAL SURFACE EQUATION WITH MULTIPLE VARIABLE EXPONENTS

. We are concerned with the existence of nontrivial weak solutions for a class of generalized minimal surface equations with subcritical growth and Dirichlet boundary condition. In relationship with the values of several variable exponents, we establish two suﬃcient conditions for the existence of solutions. In the ﬁrst part of this paper, we prove the existence of a non-negative solution. Next, we are concerned with the existence of inﬁnitely many solutions in a symmetric abstract setting.

1. Introduction and abstract setting. The Plateau problem is associated with the study of minimal surfaces. This problem consists in finding a surface with least area in R 2 that spans a given closed curve with smooth boundary. Such a surface is described by solutions of the following minimal surface equation div ∇u This problem was formulated by J.L. Lagrange in 1760 and it is named after the Belgian physicist J. Plateau who experimented with soap films. The concept of mean curvature of a surface was used by J.B. Meusnier in 1776 in his works on minimal surfaces related with the Plateau problem. These problems have been intensively studied in the last few decades, see, e.g., E. Giusti [12], B. Kawohl [14], and M. Struwe [23].
The differential operator in Eq. (1) has been extended into several directions. In the present paper we are interested in a nonlinear problem driven by the generalized mean curvature operator div (1 + |∇u| 2 ) (p(x)−2)/2 ∇u .
We refer to Example 5 in the monograph of V. Rȃdulescu and D. Repovš [22, p. 28].
The main results in this paper are concerned with the existence of nontrivial solutions for a nonhomogeneous perturbation of the differential operator div (1 + |∇u| 2 ) (p(x)−2)/2 ∇u and corresponding to a power-type reaction term with variable exponent. More precisely, we study the following nonlinear problem −div (1 + |∇u| 2 ) (p(x)−2)/2 ∇u + |u| q(x)−2 u = λ|u| r(x)−2 u in Ω u = 0 on ∂Ω, where Ω ⊂ R N (N ≥ 2) is a smooth bounded domain and λ is a positive parameter.
In accordance with the values of the variable exponents p, q and r, we study two different situations in a subcritical abstract setting. In the first case, we establish the existence of solutions for small perturbations of the reaction term, that is, provided that λ > 0 is small enough. In the second case, we prove that nontrivial solutions do exist for all λ > 0. The proofs rely on variational arguments and essential tools are Ekeland's variational principle [9] and the symmetric mountain pass theorem of A. Ambrosetti and P. Rabinowitz [3].
This paper is organized as follows. In the next section, we recall some basic properties of the Lebesgue and Sobolev spaces with variable exponent. The main results are stated in section 3 while the proofs are developed in sections 4 and 5 of this paper.
Notation. for given real numbers a and b, we denote a ∧ b := min{a, b} and a ∨ b := max{a, b}.
For two functions f, g : D → R we write 2. Function spaces with variable exponent. In this section we recall some basic facts that concern the Lebesgue and Sobolev function spaces with variable exponent. We refer to V. Rȃdulescu and D. Repovš [22] for proofs and additional results.
Let  For any p ∈ C + (Ω), we define the variable exponent Lebesgue space Then L p(x) (Ω) is a Banach space endowed with the Luxemburg norm, namely The function space L p(x) (Ω) is reflexive for all p ∈ C and continuous functions with compact support are dense in L p(x) (Ω). The inclusion between Lebesgue spaces also generalizes the classical framework, namely if p 1 , p 2 are variable exponents so that p 1 ≤ p 2 in Ω then there exists the continuous embedding L p2(x) (Ω) → L p1(x) (Ω).
Let L p (x) (Ω) be the conjugate space of L p(x) (Ω), where 1/p(x) + 1/p (x) = 1. For any u ∈ L p(x) (Ω) and v ∈ L p (x) (Ω) the following Hölder-type inequality holds: Consider the mapping ρ p(x) : If (u n ), u ∈ L p(x) (Ω) then the following relations are true: Let W 1,p(x) (Ω) denote the variable exponent Sobolev space defined by On W 1,p(x) (Ω) we may consider one of the following equivalent norms (Ω) as the closure of the set of compactly supported W 1,p(x)functions with respect to the norm u p(x) . Equivalently, we can also use the closure (Ω) can be also defined, in an equivalent manner, as the closure of C ∞ 0 (Ω) with respect to the norm u p(x) = |∇u| p(x) .

RAMZI ALSAEDI
The space (W 1,p(x) 0 (Ω), · ) is a separable and reflexive Banach space. Moreover, if p 1 , p 2 are variable exponents so that p 1 ≤ p 2 in Ω then there exists the continuous embedding W (Ω) then the following properties are true: Let p * : Ω → R be the critical Sobolev function associated to p ∈ C, namely We recall that if p, q ∈ C and q(x) < p * (x) for all x ∈ Ω then the embedding W 3. Main results. Throughout this paper, we are interested in combined effects generated by the competition between the growth of the three variable exponents involved in the generalized mean curvature problem (2). We consider problem (2) under different assumptions concerning the values of p, q and r and we establish sufficient conditions for the existence of nontrivial weak solutions. In both cases considered in the present paper, we are concerned with a subcritical abstract setting. The first main result is concerned with the existence of a non-negative solution while the second theorem established in this paper deals with the existence of infinitely many solutions. We say that ∈ W We are first concerned with the existence of non-negative solutions of problem (2). More precisely, we study the following nonlinear problem In order to establish a sufficient condition for the existence of solutions of problem (11), we assume that p ∈ C and q, r ∈ C + (Ω) satisfy the hypothesis Under this assumption, we establish the existence of a non-negative solution in the case of small perturbations of the reaction term, namely if the positive parameter λ is small enough. A related result has been established in R. Alsaedi [1, Theorem 3.1].
In particular, under hypothesis (12), we deduce that A counterpart of Theorem 3.1 in the semilinear elliptic case includes problems of the type where 1 < r < 2. We refer to H. Brezis and L. Oswald [5] for a thorough analysis of problem (13). In [5], the existence of the (unique) nontrivial solution of problem (13) relies on a minimization technique. In addition, it is observed that the energy functional to be minimized associated to problem (13) is convex with respect to the variable ρ = u 2 . We notice that problem (13) has a solution for all λ > 0. However, in our anisotropic case described in Theorem 3.1, we are not able to prove that the solution exists for all λ > 0. This comes essentially from the combined growths of the variable exponents p, q and r, as they are described in hypothesis (12). In problem (11) the presence of variable exponents allow us to establish the existence of solutions only for small values of λ. At this stage, we are not able to say what happens for values of λ larger than λ * . This is mainly due to the fact that in our case described by hypothesis (12), we cannot apply coercivity arguments for the energy functional associated to problem (11). Next, we are interested in the existence of infinitely many solutions of problem (2). The main assumption is p + ∨ q + < r − and r p * .
This hypothesis corresponds to the case where the subcritical reaction term dominates the left-hand side of problem (2). In this case, the associated energy is not coercive but has a mountain pass geometry. The symmetry of the problem implies the existence of infinitely many solutions. However, we cannot assert that these solutions have constant sign or if they are nodal.

A counterpart of Theorem 3.2 in the semilinear elliptic case includes the problem
where 2 < r < 2N/(N − 2). This problem goes back to A. Ambrosetti and P. Rabinowitz [3] who used the symmetric version of their mountain pass theorem in order to prove the existence of infinitely many solutions for all λ > 0. In both cases described by Theorems 3.1 and 3.2, the associated energy is J (u) : W 1,p(x) 0 (Ω) → R defined by Any of the hypotheses (12) or (14) implies that W

1,p(x) 0
(Ω) is compactly embedded both into L q(x) (Ω) and L r(x) (Ω), hence J is well defined. Moreover, J is of class C 1 and its directional derivative is 4. Proof of Theorem 3.1. Under hypothesis (12), we show that the energy functional J satisfies only one of the geometric assumptions of the mountain pass theorem. In our arguments we apply some ideas found in the proof of Theorem 2.1 in [17].
By hypothesis (12) and compact embeddings for Lebesgue and Sobolev spaces with variable exponent, there exists a positive constant C 0 such that for all u ∈ W 1,p(x) 0 (Ω) We divide the proof into several steps.
Step 3. Existence of almost critical points.
Steps 1 and 2 show that there exist λ * > 0 and R 0 > 0 such that for all λ ∈ (0, λ * ) Fix λ ∈ (0, λ * ) and We apply the Ekeland variational principle to J restricted to the complete met- (Ω). For any ε > 0 as above we find u ε such that With a standard argument, relation (20) shows that J (u ε ) ≤ ε. Thus, using (19), we conclude that u ε is an "almost critical point" of J for all ε > 0 sufficiently small. Proof of Theorem 3.1 completed. We know that there exists a bounded sequence (u n ) ⊂ W 1,p(x) 0 (Ω) (with u n ≤ R 0 for all n) such that By reflexivity and compact embeddings, we can assume that, up to a subsequence, Relation (21) implies that We first argue that Ω |u n | q(x)−2 u n (u n − u)dx → 0 as n → ∞.

RAMZI ALSAEDI
With a similar argument we deduce that Ω |u n | r(x)−2 u n (u n − u)dx → 0 as n → ∞.
With the same arguments as in the proof of Theorem 3.1 of X. Fan and Q. Zhang [10] we deduce that L is continuous, bounded, strictly monotone, and a mapping of type (S + ), that is, if u n u and lim sup n→∞ Lu n − Lu, u n − u ≤ 0, (Ω). In our case, using relation (25), we conclude that u n strongly converges to u in W 1,p(x) 0 (Ω). We deduce that J (u) = m < 0 and J (u) = 0, hence u is a nontrivial solution of problem (2).

Proof of Theorem 3.2.
We check the hypotheses of the symmetric mountain pass theorem of A. Ambrosetti and P. Rabinowitz [3,Corollary 2.9]. For the convenience of the reader, we recall this fundamental result in what follows.
Theorem 5.1. Let E be an infinite dimensional Banach space over R. We assume that I is an even functional of class C 1 which satisfies the following hypotheses: (I 1 ) there exist positive numbers R 0 and a 0 such that I(u) ≥ a 0 for all u ∈ E with u = R 0 ; (I 2 ) if (u n ) ⊂ E with I(u n ) bounded and I (u n ) → 0, then (u n ) possesses a convergent subsequence; (I 3 ) for any finite dimensional subspace F of E, the set P := {u ∈ F ; I(u) ≥ 0} is bounded.
Then I has infinitely many distinct pairs of critical points.
We first prove the existence of a "chain of mountains" near the origin for the associated energy functional J . Next, we prove that if F ⊂ W 1,p(x) 0 (Ω) is an arbitrary finite dimensional subspace then the set of points u ∈ F with J (u) ≥ 0 is bounded. Finally, we prove that the even functional J satisfies the Palais-Smale compactness condition.
We have Let F be an arbitrary finite dimensional subspace of W 1,p(x) 0 (Ω). Then Ω |u| r − dx 1/r − is a norm in F , which is equivalent with the norm u p(x) . Thus, there exists C 6 > 0 such that |u| r − ≥ C 6 u p(x) for all u ∈ F.