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

In this paper, we introduce the s(., .)-fractional Musielak–Sobolev spaces Ws(x,y)LΦx,y(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W^{s(x,y)}L_{\varPhi _{x,y}}(\Omega )$$\end{document}. Then, we show that there exists λ∗>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _*>0$$\end{document} such that any λ∈(0,λ∗)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \in (0, \lambda _*)$$\end{document} is an eigenvalue for the following problem, by means of Ekeland’s variational principle (Pa)-Δa(x,.)s(x,.)u=λ|u|q(x)-2uinΩ,u=0inRN\Ω,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} ({\mathcal {P}}_a) \left\{ \begin{array}{clclc} \left( -\Delta \right) ^{s(x,.)}_{a_{(x,.)}} u &{} = &{} \lambda |u|^{q(x)-2}u &{} \text { in }&{} \Omega , \\ \\ u &{} = &{} 0 \hspace{0.2cm} &{} \text { in } &{} {\mathbb {R}} ^N\setminus \Omega , \end{array} \right. \end{aligned}$$\end{document}where Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} is a bounded open subset of RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}} ^N$$\end{document} with C0,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{0,1}$$\end{document}-regularity and bounded boundary.