Weak solutions for the fractional Kirchhoff-type problem via Young measures

Abstract

The aim of this paper is to investigate the existence of weak solutions to the following Kirchhoff-type problem:$$\begin{cases}M\left([u]_{s p}^p\right) (-\Delta)_p^s (u)=f(x,u) \quad &\text { in } \Omega,\\ u=0 \quad &\text { in } \mathbb{R}^n \backslash \Omega,\end{cases}$$where $\Omega\subset\mathbb{R}^n$, $0<s<1<p<\infty$, $[u]_{s p}$ is Gagliardo semi-norm, $M$ is a continuous function with value in $\mathbb{R}^+$, $f$ a given function and $(-\Delta)_p^s$ is the fractional $p$-Laplacian operator.Under appropriate assumptions on the main functions, weobtain the existence results by applying the Galerkin method combined with the theory of Young measures.