The fibering map approach for a singular elliptic system involving the $p(x)$-Laplacian and nonlinear boundary conditions

The purpose of this work is to study the existence and multiplicity of positive solutions for a class of singular elliptic systems involving the p(x)Laplace operator and nonlinear boundary conditions.


Introduction
This paper is concerned with the multiplicity of positive solutions for the following singular elliptic system involving the p(x)-Laplace operator and sub-linear Neumann nonlinearities: (1.1) Here Ω ⊂ R N (N ≥ 2) is a bounded domain with C 2 boundary; λ, µ are two parameters; a, b, c ∈ C(Ω) are non-negative weight functions with compact support in Ω. For any continuous and bounded function a we define a + := ess sup a(x) and a − := ess inf a(x). We assume the following on p, q, r and α: (A1) p(x), q(x), r(x) ∈ C(Ω) are such that 0 < 1 − α(x) < p(x) < q(x) + r(x) < p * (x) (where p * (x) = N p(x) N −p(x) if p(x) < N and p * (x) = ∞ if p(x) ≥ N ), and p − ≤ p + < q − + r − ≤ q + + r + .

MOUNA KRATOU AND KAMEL SAOUDI
The operator ∆ p(x) u := div(|∇u| p(x)−2 ∇u), where p is a continuous non-constant function, is called p(x)-Laplace. This differential operator is a natural generalization of the p-Laplace operator ∆ p u := div(|∇u| p−2 ∇u), where p > 1 is a real constant. However, the p(x)-Laplace operator possesses more complicated nonlinearity than the p-Laplace operator, due to the fact that ∆ p(x) is not homogeneous. This fact implies some difficulties; for example, we can not use the Lagrange multiplier theorem in many problems involving this operator.
The study of differential and partial differential equations involving variable exponent is a new topic. The interest in studying such problems was stimulated by their applications in elastic mechanics, fluid dynamics, electrorheological fluids, image processing, flow in porous media, calculus of variations, non-linear elasticity theory, heterogeneous porous media models, etc. (see [1,5]). These physical problems were facilitated by the development of Lebesgue and Sobolev spaces with variable exponent. Before giving our main results, let us briefly recall literature concerning related non-linear equations involving the p(x)-Laplace operator. The existence and multiplicity of solutions of elliptic equations with variable exponents involving the p(x)-Laplace operator have been extensively investigated using various methods, specially variational techniques, and have received much attention. In that context, we would like to mention [2,9,10,17,19,25,26,27,29] and the references therein.
The fibering map approach for describing the Nehari manifolds and seeking solutions in an appropriate subset of the Sobolev space is introduced by Drabek and Pohozaev in [6]. In variable exponent cases this method has some difficulties in comparison with the Nehari manifolds approach in p-Laplacian problems. This is due to the non-homogeneity of the variable exponent p. Nevertheless, in recent years, several authors have used the Nehari manifold and fibering maps to solve quasilinear problems with variable exponent (see [18,22,30]).
Problem (1.1) has been also studied with different elliptic operators. We refer the reader to the monograph by Ghergu and Rȃdulescu [14] for a more general presentation of these results and the survey article of Crandall, Rabinowitz, and Tartar [4]. After this, many authors have considered the problem above for Laplacian operators, p-Laplacian operators, fractional Laplacian or fractional p-Laplacian, using the technique used in [4] or a combination of this approach with Nehari's and Perron's methods; we would like to mention [3,12,13,15,24,28].
However, as far as we know, there are few results on p(x)-Laplacian systems with concave/convex nonlinearities (see [21,23] and references therein). Motivated by the above results, in the present paper we are interested in the multiplicity of solutions for the singular p(x)-Laplacian system (1.1) by using the Nehari manifold decomposition.
Here we state our main result.
where c 8 , c 9 are positive constants, such that the problem (1.1) has at least two non-negative solutions for all 0 < λ + µ < Λ 0 .
This paper is organized as follows. In Section 2, we will recall some basic facts about the variable exponent Lebesgue and Sobolev spaces which we will use later.
In Section 3, we analyze the fibering map related to the Euler functional associated to the problem (1.1). Proofs of our results will be presented in Sections 4 and 5.

Generalized Lebesgue-Sobolev spaces setting
To deal with the p(x)-Laplacian problem, we need to introduce some functional spaces L p(·) (Ω), W 1,p(·) (Ω), W 1,p(·) 0 (Ω), and some properties of the p(x)-Laplacian that we will use later. Denote by S(Ω) the set of all measurable real-valued functions defined in Ω. Note that two measurable functions are considered as the same element of S(Ω) when they are equal almost everywhere. Let where c is a measurable real-valued function and c(x) > 0 for x ∈ Ω. The space (L p(·) (Ω), | · | p(·) ) becomes a Banach space. We call it variable exponent Lebesgue space. Moreover, this space is a separable, reflexive, and uniform convex Banach space; see [11, Theorems 1.6, 1.10, 1.14]. The variable exponent Sobolev space can be equipped with the norm u = |u| p(·) + |∇u| p(·) , for all u ∈ W 1,p(·) (Ω).

Theorem 2.2 (See [8, 16]). Let Ω ⊂ R N be an open bounded domain with Lipschitz boundary and assume that
for all x ∈ Ω, then there exists a continuous embedding The following three theorems play an important role in the present paper.  [20]). Assume that the boundary of Ω possesses the cone property and p ∈ C(Ω).

Fibering map analysis for problem (1.1)
In what follows, W will denote the Cartesian product of two Sobolev spaces , which is equivalent to the standard one.
Associated to the problem (1.1) we define the functional E λ,µ : W → R given by and Note that using a new cut-off functional (see [29,Lemma A.3]) allows us to apply the variational method. Precisely, we obtain the C 1 -differentiability of the associated cut-off functional. and In many problems, such as problem (1.1), E λ,µ is not bounded below on the whole space W , but is bounded below on the corresponding Nehari manifold, which is defined by It is clear that all critical points of E λ,µ must lie on N λ,µ . We will see below that local minimizers of E λ,µ on N λ,µ are usually critical points of E λ,µ . Then, it is easy to see that u ∈ N λ if and only if We note that N λ,µ contains every solution of problem (1.1). Then, for (u, v) ∈ N λ,µ we have Now, we know that the Nehari manifold is closely related to the behavior of the Such maps are called fiber maps and were introduced by Drabek and Pohozaev in [6]. and Proof. The result is a consequence of the fact that From Lemma 3.2, we have that the elements in N λ,µ correspond to stationary points of the maps Φ u,v (t) and, in particular, (u, v) ∈ N λ,µ if and only if Φ u,v (1) = 0. Hence, it is natural to split N λ,µ into three parts corresponding to local minima, local maxima, and points of inflection Φ u,v (t) defined as follows: Our first result is the following. Proof. Suppose that (u, v) ∈ N λ,µ and (u, v) > 1. Without loss of generality, we may assume that u p(x) , u q(x) > 1. Therefore, using (2.2)-(2.4) and (3.1) we estimate E λ,µ (u, v) as follows: This implies that E λ,µ is coercive and bounded below.

Lemma 3.4. Let (u, v) be a local minimizer for E λ,µ on subsets
applying the theory of Lagrange multipliers we get the existence of σ ∈ R such that E λ,µ (u, v) = σI λ,µ (u, v).

Existence of minimizer on N + λ,µ
In this section, we will show that the minimum of E λ,µ is achieved in N + λ,µ . Also, we show that this minimizer is also the first solution of problem (1.1). (4.1)

Existence of minimizer on N − λ,µ
In this section, we shall show the existence of a second solution of problem (1.1) by proving the existence of a minimizer of E λ,µ on N − λ,µ .

THE FIBERING MAP APPROACH FOR A SINGULAR NEUMANN SYSTEM 185
On the other hand, from the definition of the E λ , we can write Therefore, using (5.1), (5.2), and (3.1) we get infimum is zero, then from (5.3), the minimizing sequence {(u k , v k )} converges strongly in W to (0, 0) and (0, 0) / ∈ N − λ,µ , a contradiction. Therefore c − λ,µ > 0 follows from the definition of c − λ,µ , and the proof is complete.
Theorem 5.2. If λ, µ are such that 0 < λ + µ < Λ 0 , then the functional I λ,µ has a minimizer ( Proof. Since E λ,µ is bounded below on N λ,µ , and so on N − λ,µ , there exists a sequence Since E λ,µ is coercive, {(u n , v n )} is bounded in W . Thus, we may assume that, without loss of generality, weakly in W , and by the compact embedding we have (Ω).