Existence and multiplicity of solutions for a fractional $p$-Laplacian problem of Kirchhoff type via Krasnoselskii's genus

summary:We use the genus theory to prove the existence and multiplicity of solutions for the fractional $p$-Kirchhoff problem $$ \begin {cases} \displaystyle -\biggl [M \biggl (\int _{Q}\dfrac {\vert u(x)-u(y)\vert ^{p}}{\vert x-y \vert ^{N+ps}} {\rm d}x {\rm d}y\biggr )\biggr ]^{p-1} (-\Delta )_{p}^{s}u=\lambda h(x,u) \quad \text {in}\ \Omega , \\ u=0 \quad \text {on}\ \mathbb {R}^N \setminus \Omega , \end {cases} $$ where $\Omega $ is an open bounded smooth domain of $\mathbb {R}^N$, $p>1$, $N>ps$ with $s\in (0,1)$ fixed, $Q = \mathbb {R}^{2N}\setminus (C\Omega \times C\Omega )$, $\lambda > 0$ is a numerical parameter, $M$ and $h$ are continuous functions