Existence and multiplicity of solutions for Schr\"odinger-Kirchhoff-type equations involving the fractional $p(x,\cdot)$-Laplacian without the (AR) condition

Fractional $p(x,\cdot)$-Kirchhoff type equation without the (AR) condition

Abstract

The purpose of this paper is to investigate the existence and multiplicity of weak solutions for a Kirchhoff-type problems driven by the non-local integro-differential operator of elliptic type
$$
\left\{\begin{array}{ll}
M \big(\sigma_{p(x,y)}(u)\big)\L_K^{p(x,\cdot)} (u) =f(x,u) &
\text{ in } \Omega,\\
u(x)=0 & \textrm{ in } \R^{N}\backslash \Omega,
\end{array}\right.
$$
where
$$
\sigma_{p(x,y)}(u)=\int _{\Q} \frac{|u(x)-u(y)|^{p(x,y)}}{p(x,y)}K(x,y)\,dx\,dy,
$$
$\L_{K}^{p(x,\cdot)}$ is a non-local operator with singular kernel $K$, $\Omega$ is an open bounded subset of $\R^N$ with Lipschitz boundary $\partial \Omega$, $M$ is a continuous function and $f$ is a Carath\'{e}odory function. Under suitable assumptions on $f(x,u)$ without (AR) condition, the existence and multiplicity solutions for the problem is obtained by using the Mountain Pass Theorem and the Fountain Theorem.