Multiplicity results for nonlocal fractional $p$-Kirchhoff equations via Morse theory

Abstract

In this paper, we apply Morse theory and local linking to study the existence of nontrivial solutions for Kirchhoff type equations involving the nonlocal fractional $p$-Laplacian with homogeneous Dirichlet boundary conditions: \[ \begin{cases} \bigg[M\bigg(\displaystyle\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\,dxdy\bigg)\bigg]^{p-1} (-\Delta)_p^su(x)=f(x,u)&\mbox{in }\Omega,\\ u=0&\mbox{in } \mathbb{R}^{N}\setminus\Omega, \end{cases} \] where $\Omega$ is a smooth bounded domain of $\mathbb{R}^N$, $(-\Delta)_p^s$ is the fractional $p$-Laplace operator with $0<s<1<p<\infty$ with $sp<N$, $M \colon \mathbb{R}^{+}_{0}\rightarrow \mathbb{R}^{+}$ is a continuous and positive function not necessarily satisfying the increasing condition and $f$ is a Carathéodory function satisfying some extra assumptions.