Existence of solutions for a nonlocal Kirchhoff type problem in Fractional Orlicz-Sobolev spaces

Abstract

In this paper, we investigate the existence of weak solution for a Kirchhoff type problem driven by a nonlocal operator of elliptic type in a fractional Orlicz-Sobolev space, with homogeneous Dirichlet boundary conditions {\small$$ (D_{K,A}) \hspace*{0.5cm} \left\{ \begin{array}{clclc} M\left( \displaystyle \int_{\R^{2N}}A\left( [u(x)-u(y)] K(x,y)\right) dxdy\right) \mathcal{L}^K_A u & = & f(x,u) & \text{ in }& Ω, \hspace*{7cm} u & = & 0 \hspace*{0.2cm} \hspace*{0.2cm} & \text{ in } & \R^N\setminus Ω. \label{eq1} \end{array} \right. $$ } Where ${\mathcal{L}}_A^K$ is a nonlocal operator with singular kernel $K$ and $A$ is an $N$-function, $\Omega$ is an open bounded subset in ${\mathbb{R}}^N$ with Lipschitz boundary $\partial \Omega$.