Fractional Sobolev space with variable exponents: Study of Kirchhoff problem by Berkovits degree theory

Abstract

This paper discusses a class of This paper discusses a class of Kirchhoff boundary value problem  that involve the fractional $\tau(\cdot)$-Laplacian operator of the form\begin{equation*} \begin{cases} \Big(\alpha+\beta[u]_{s,\tau}\Big)\left(-\Delta\right)_{\tau(x,\cdot)}^{s} u(x)+|u(x)|^{r(x)-2} u(x)=\xi|u(x)|^{\mu(x)-2} u(x) & \text { in } \Omega, \\ u=0 & \text { in } \mathbb{R}^{N} \backslash \Omega . \end{cases}\end{equation*}where $\om$ is a smooth bounded domain in $\R^N$, $(N\geq2)$, $0<s<1$, $\xi$ is a positive parameter.Utilizing Berkovits degree theory with thetheory of fractional Sobolev spaces with variable exponents, we demonstrate the existence of nontrivial weak solutions for this problem.