Eigenvalue type problem in $s(.,.)$-fractional Musielak-Sobolev spaces

Abstract

In this paper, first we introduce the $s(.,.)$-fractional Musielak-Sobolev spaces $W^{s(x,y)}L_{\varPhi_{x,y}}(\Omega)$. Next, by means of Ekeland's variational principal, we show that there exists $\lambda_*>0$ such that any $\lambda\in(0, \lambda_*)$ is an eigenvalue for the following problem $$(\mathcal{P}_a) \left\{ \begin{array}{ll}\left( -\Delta\right)^{s(x,.)}_{a_{(x,.)}} u = \lambda |u|^{q(x)-2}u &\quad {\rm in}\ \Omega, \\ \qquad\quad u = 0 &\quad {\rm in }\ \mathbb{R}^N\setminus \Omega, \end{array} \right. $$ where $\Omega$ is a bounded open subset of $\mathbb{R}^N$ with $C^{0,1}$-regularity and bounded boundary.