Ekeland’s Variational Principle for the Fractional p(x)-Laplacian Operator

Abstract

In this paper, using adequate variational techniques, mainly based on Ekeland’s variational principle, we will establish the existence of a continuous family of eigenvalues for problems driven by the fractional p(x)-Laplacian operator $(-\varDelta _{p(x)})^{s}$, with homogenous Dirichlet boundary conditions. More precisely, we show that there exists $\lambda ^{*}>0$, such that for all $\lambda \in (0,\lambda ^{*})$ is an eigenvalue of the problem

$$(\mathscr {P}_{ s})~~ \left\{ \begin{array}{llllll} \ (-\varDelta _{p(x)})^{s}u(x) &{} &{} =\lambda |u(x)|^{r(x)-2}u(x)&{}in&{} ~\varOmega ,\\ \\ \qquad \qquad \qquad u&{}&{}=0 &{}in&{} ~\mathbb {R}^{N}\setminus \varOmega , \end{array} \right. $$

where $\varOmega$ is a bounded open set of $\mathbb {R}^{N}$, $\lambda >0$, p and r are a continuous variable exponents.