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

Abstract

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

arXiv admin note: text overlap with arXiv:2203.01756, arXiv:2007.11043