Topological degree methods for non-local fractional $\mathfrak{p}(\mathrm{x}, .)$- laplacian elliptic equation

Résumé

The purpose of this paper is mainly to investigate the existence of weak solutions to a class of fractional $\mathfrak{p}(\mathrm{x}, .)$- Laplacian problem as follows: \begin{eqnarray}\label{k1} %\begin{gathered} \left\{\begin{array}{llll} (\mathcal{L}_{\mathrm{K}} ^{s})_{\mathfrak{p}(\mathrm{x}, .)} \mathrm{v(\mathrm{x})}& = \lambda \beta(\mathrm{x})\vert \mathrm{v(\mathrm{x})}\vert^{r(\mathrm{x})-2}\mathrm{v(\mathrm{x})} + f(\mathrm{x}, \mathrm{v(\mathrm{x})})& \text { in }& U, \\ \hspace{1.6cm} \mathrm{v}&=0 & \text { in }& \mathbb{R}^{N} \backslash U, \end{array}\right. %\end{gathered} \end{eqnarray} where $f:U\times \mathbb{R} \rightarrow \mathbb{R}$ is a Carath\'{e}odory function, $U$ is a smooth bounded domain of $\mathbb{R^{N}}$, and $(\mathcal{L}_{\mathrm{K}} ^{s})_{\mathfrak{p}(\mathrm{x}, .)} $ is the fractional $\mathfrak{p}(\mathrm{x}, .)$- Laplacian operator. The main tool used here is the topological method introduced by Berkovits for demi-continuous operators of generalized type $(S_{+}),$ which is based on transforming our problem to an abstract Hammerstein equation.