A critical fractional <i>p</i>-Kirchhoff type problem involving discontinuous nonlinearity

Abstract

The aim of this paper is to discuss the existence and multiplicity of solutions for the following fractional \begin{document}$p$\end{document} -Kirchhoff type problem involving the critical Sobolev exponent and discontinuous nonlinearity: \begin{document}$\begin{align*}M\left(\displaystyle\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}dxdy\right)(-\Delta)_p^su = \lambda|u|^{p_s^*-2}u+f(x,u)~~\mbox{in }\,\,\mathbb{R}^N,\end{align*}$ \end{document} where \begin{document}$M(t) = a+bt^{\theta-1}$\end{document} for \begin{document}$t\geq 0$\end{document} , \begin{document}$a\geq 0, b>0,\theta>1$\end{document} , \begin{document}$(-\Delta)_p^s$\end{document} is the fractional \begin{document}$p$\end{document} --Laplacian with \begin{document}$0 and \begin{document}$1 , \begin{document}$p_s^* = Np/(N-ps)$\end{document} is the critical Sobolev exponent, \begin{document}$\lambda>0$\end{document} is a parameter, and \begin{document}$f:\mathbb{R}^N\times\mathbb{R}\rightarrow\mathbb{R}$\end{document} is a function. Under suitable assumptions on \begin{document}$f$\end{document} , we show that there exists \begin{document}$\lambda_0>0$\end{document} such that the above equation admits at least one nontrivial nonnegative solution provided \begin{document}$\lambda by using the nonsmooth critical point theory for locally Lipschitz functionals. Furthermore, for any \begin{document}$k\in\mathbb{N}$\end{document} , there exists \begin{document}$\Lambda_k>0$\end{document} such that the above equation has \begin{document}$k$\end{document} pairs of nontrivial solutions if \begin{document}$\lambda . The main feature is that our paper covers the degenerate case, that is the coefficient of \begin{document}$(-\Delta)_p^s$\end{document} may be zero at zero. Moreover, the existence results are obtained when \begin{document}$f$\end{document} is discontinuous. Thus, our results are new even in the semilinear case.