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A critical fractional p-Kirchhoff type problem involving discontinuous nonlinearity

  • * Corresponding author: Binlin Zhang

    * Corresponding author: Binlin Zhang

M. Xiang was supported by the National Natural Science Foundation of China (No. 11601515) and the Fundamental Research Funds for the Central Universities (No. 3122017080). B. Zhang was supported by Natural Science Foundation of Heilongjiang Province of China (No. A201306) and Research Foundation of Heilongjiang Educational Committee (No. 12541667)

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  • 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.

    Mathematics Subject Classification: Primary: 35R11, 35A15; Secondary: 47G20.


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