On a class of fractional Laplacian problems with variable exponents and indefinite weights

Abstract

Let $\Omega \subset {\mathbb {R}}^N, N\ge 2$, be a bounded smooth domain. In this paper, we consider a class of fractional Laplacian problems of the form

$$\begin{aligned} \left\{ \begin{array}{ll} (\Delta )^s_{p_1(x,.)}u(x)+(\Delta )^s_{p_2(x,.)}u(x) + |u|^{q(x)-2}u = \lambda V_1(x)|u(x)|^{r_1(x)-2}u(x) \\ \qquad - \mu V_2(x)|u(x)|^{r_2(x)-2}u(x) \hbox { in }\Omega , \\ u(x) = 0 \; \hbox { in } \partial \Omega , \end{array} \right. \end{aligned}$$

where $(\Delta )^s_{p_i(.,.)} (0<s<1), i=1,2$, are the fractional $p_i(.,.)$-Laplacians, $p_i\in C({\overline{\Omega }} \times {\overline{\Omega }}), q, r_i\in C({\overline{\Omega }}), i=1,2$ while $\lambda , \mu$ are two positive parameters, $V_1, V_2$ are weight functions in generalized Lebesgue spaces $L^{\alpha _1(.)}(\Omega )$ and $L^{\alpha _2(.)}(\Omega )$ respectively such that $V_1$ may change sign in $\Omega$ and $V_2(x)\ge 0$ for all $x\in \Omega$. Using variational techniques and Ekeland’s variational principle, we establish some existence results for the problem in an appropriate space of functions.