A critical fractional p-Kirchhoff type problem involving discontinuous nonlinearity

The aim of this paper is to discuss the existence and multiplicity of solutions for the following fractional $p$-Kirchhoff type problem involving the critical Sobolev exponent and discontinuous nonlinearity:

$\begin{align*}M\left(\displaystyle\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}dxdy\right)(-\Delta)_p^su = \lambda|u|^{p_s^*-2}u+f(x,u)~~\mbox{in }\,\,\mathbb{R}^N,\end{align*}$

where $M(t) = a+bt^{\theta-1}$ for $t\geq 0$, $a\geq 0, b>0,\theta>1$, $(-\Delta)_p^s$ is the fractional $p$--Laplacian with $0<s<1$ and $1<p<N/s$, $p_s^* = Np/(N-ps)$ is the critical Sobolev exponent, $\lambda>0$ is a parameter, and $f:\mathbb{R}^N\times\mathbb{R}\rightarrow\mathbb{R}$ is a function. Under suitable assumptions on $f$, we show that there exists $\lambda_0>0$ such that the above equation admits at least one nontrivial nonnegative solution provided $\lambda<\lambda_0$ by using the nonsmooth critical point theory for locally Lipschitz functionals. Furthermore, for any $k\in\mathbb{N}$, there exists $\Lambda_k>0$ such that the above equation has $k$ pairs of nontrivial solutions if $\lambda<\Lambda_k$. The main feature is that our paper covers the degenerate case, that is the coefficient of $(-\Delta)_p^s$ may be zero at zero. Moreover, the existence results are obtained when $f$ is discontinuous. Thus, our results are new even in the semilinear case.