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

In this paper, we introduce the s(., .)-fractional Musielak–Sobolev spaces \(W^{s(x,y)}L_{\varPhi _{x,y}}(\Omega )\) $W^{s(x,y)}{L}_{{\Phi }_{x,y}}(\mathrm{\Omega })$ . Then, we show that there exists \(\lambda _*>0\) ${\lambda }_{\ast }>0$ such that any \(\lambda \in (0, \lambda _*)\) $\lambda \in (0,{\lambda }_{\ast })$ is an eigenvalue for the following problem, by means of Ekeland’s variational principle \(\begin{aligned} ({\mathcal {P}}_a) \left\{ \begin{array}{clclc} \left( -\Delta \right) ^{s(x,.)}_{a_{(x,.)}} u &{} = &{} \lambda |u|^{q(x)-2}u &{} \text { in }&{} \Omega , \\ \\ u &{} = &{} 0 \hspace{0.2cm} &{} \text { in } &{} {\mathbb {R}} ^N\setminus \Omega , \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill ({\mathcal{P}}_a)\left\{\begin{array}{ccccc}{\left(-,\mathrm{\Delta }\right)}_{a_{(x,.)}}^{s(x,.)}u& =\hfill & {\lambda |u|}^{q(x)-2}u& \text{in}\hfill & \mathrm{\Omega },\\ \\ u& =\hfill & 0& \text{in}\hfill & {\mathbb{R}}^N\backslash \mathrm{\Omega },\end{array}\right)\end{array}$$ where \(\Omega \) $\mathrm{\Omega }$ is a bounded open subset of \({\mathbb {R}} ^N\) ${\mathbb{R}}^N$ with \(C^{0,1}\) $C^{0,1}$ -regularity and bounded boundary.