NONLOCAL EIGENVALUE PROBLEMS WITH INDEFINITE WEIGHT

In the present paper, we consider a class of eigenvalue problems driven by a nonlocal integro-di erential operator \scrL K with Dirichlet boundary conditions. Under certain assumptions on p and q, we establish that any \lambda > 0 su ciently small is an eigenvalue of the nonhomogeneous nonlocal problem (\scrP \lambda ). ®§£«ï¤ õâìáï a« á á ̄¥aâà «ì­ ̈å § ¤ ç,  ̄®¢'ï§ ­ ̈å ÷§ ­¥«®a «ì­ ̈¬ ÷­â¥£à®¤ ̈ä¥à¥­æ÷ «ì­ ̈¬ ® ̄¥à â®à®¬ \scrL K ÷§ aà ©®¢®î ã¬®¢®î  ̈à ̈å«¥.    ̄¥¢­ ̈å ̄à ̈ ̄ãé¥­ì é®¤® p ÷ q ¤®¢¥¤¥­®, é® a®¦­¥ ¤®áâ ­ì® ¬ «¥ \lambda > 0 õ ¢« á­ ̈¬ §­ ç¥­­ï¬ ­¥®¤­®à÷¤­®ù ­¥«®a «ì­®ù § ¤ ç÷ (\scrP \lambda ).

In the [15], the authors investigated the following Br ezis-Nirenberg type problem: they proved the existence of one weak solution of (1.8) through direct minimization of the energy in a small ball of a certain fractional Sobolev space. Also, in [23], Nguyen Thanh Chung considered the following problem where \scrL p(x,y) is the fractional p(x, y)-Laplace operator given by (1.10) where p.v. is a commonly used abbreviation in the principal value sense. He established some results on the existence of a continuous family of eigenvalues using variational techniques and Ekeland's variational principle.
Note that the operator \bigr) s is the fractional version of well known p(x)-Laplacian operator \Delta p(x) u(x) = \mathrm{ \mathrm{ \mathrm{ . On the other hand, we remark that in the constant exponent case it is known as the fractional p -Laplacian operator ( - \Delta p ) s . This nonlocal nonlinear operator is consistent, up to some normalization constant depending upon N and s, with the linear fractional Laplacian ( - \Delta ) s in the case p = 2. The interest for this last operator and more generally pseudo-dierential operators has constantly increased over the last few years, although such operators have been a classical topic of functional analysis since long ago. Nonlocal operators such as ( - \Delta ) s and its generalisation \scrL K like in probleme (1.8) (for more details see [26,27,28]) naturally arise in continuum mechanics, phase transition phenomena, population dynamics and game theory, as they are the typical outcome of stochastical stabilization of L evy processes, see e.g. [6,21,22]. The interest in studying non-local integro-dierential was stimulated by their applications. Indeed, they have impressive applications in dierent elds, as the thin obstacle problem, optimization, nance, stratied materials, anomalous diusion, crystal dislocation, deblurring and denoising of images, and so on. For further details we refer to [7,8,9,10,20,24] and the references therein. Now, we introduce the nonlocal integro-dierential operator of elliptic type \scrL p(x) K which generalizes ( - \Delta p ) s , for any xed s \in (0, 1), as follows: where p : \BbbR N \times \BbbR N -\rightar (1, +\infty ) is a continuous bounded function satifying (1.2), (1.3) and p((x, y) -(z, z)) = p(x, y) \forall (x, y), (z, z) \in \BbbR N \times \BbbR N , (1.11) The kernel K : \BbbR N \times \BbbR N -\rightar (0, +\infty ) is a measurable function with the following properties: K(x, y) = K(y, x) \forall (x, y) \in \BbbR N \times \BbbR N (1.12) and \exists k 0 > 0 such that (1.14) A typical example for K is given by the singular s . We will introduce the functional space which was introduced in [2] by Benkirane et al., we give the general fractional Sobolev space with variable exponent as follows where \Omega be an open bounded subset of \BbbR N and Q dened by Q := \BbbR N \times \BbbR N \setminu (\scrC \Omega \times \scrC \Omega ), with \scrC \Omega = \BbbR N \setminu \Omega . Note that, the space W K,p(x,y) (\Omega ) is a Banach space (see [2] ) and endowed with the norm The space (X, \| \cdot \| X ) is separable and uniformly convex reexive, see [2].
In this paper, we are inspired by the results on the p(x)-Laplacian problems with weight introduced in [1, 5, 18] and some results on the theory of fractional Sobolev spaces with variable exponent due to Kaufmann et al. [17] and Bahrouni et al. [4]. The aim of this paper is to investigate problem (1.1) by adapting the variational techniques. We will study a class of eigenvalue problems with indenite weight for fractional p(x, y)-Laplacian equations and we establish that any \lambda > 0 suciently small is an eigenvalue of the above nonhomogeneous nonlocal problem. The proof relies on some variational arguments based on Ekeland's variational principle. The main result of the present paper reads as follows: (1.14).
Then there exists \lambda \star > 0 such that for all \lambda \in (0, \lambda \star ) is an eigenvalue of problem (\scrP K ).
The rest of this paper is structured as follows. Section 2 states some preliminary properties to establish our results presented in Section 3. In section 3, we establish and prove our main theorem.

Preliminaries and technical lemmas
In this section, we recall some denitions and some properties about generalized Lebesgue spaces L r(x) (\Omega ) and fractional Sobolev spaces with variable exponent, which we will use later (For more details see [2,13,14,19]). Dene the generalized Lebesgue space by: For all u \in L r(x) (\Omega ) and v \in L \r(x) (\Omega ) the H\" older's type inequality holds true.
On the other hand, for any u \in W K,p(x,y) 0 (\Omega ), we dene the functional is the the dual of X 0 = W K,p(x,y) 0 (\Omega ).
In the following Lemma, we introduce fundamental properties of the operator \scrL p(

Main result
We begin by the following denition. Denition 3.1. We say that u \in X 0 is a weak solution of problem (\scrP K ), if for all Moreover, we say that \lambda is an eigenvalue of problem (\scrP K ), if there exists u \in X 0 \setminu \{ 0\} which satises (3.18), i.e. u is the corresponding eigenfunction to \lambda .
Proof of Lemma 3.3: Using the same argument as in ( [23], see Theorem 3.2).
Proof of Lemma 3.4: Using the same argument as in ( [23], see Theorem 3.3).