Critical Kirchhoff-type equation with singular potential

Abstract

In this paper, we deal with the following Kirchhoff-type equation: \begin{equation*} -\bigg(1 +\int_{\mathbb{R}^{3}}|\nabla u|^{2}dx\bigg) \Delta u +\frac{A}{|x|^{\alpha}}u =f(u),\quad x\in\mathbb{R}^{3}, \end{equation*} where $A> 0$ is a real parameter and $\alpha\in(0,1)\cup ({4}/{3},2)$. Remark that $f(u)=|u|^{2_{\alpha}^{*}-2}u +\lambda|u|^{q-2}u +|u|^{4}u$, where $\lambda> 0$, $q\in(2_{\alpha}^{*},6)$, $2_{\alpha}^{*}=2+{4\alpha}/({4-\alpha})$ is the embedding bottom index, and $6$ is the embedding top index and Sobolev critical exponent. We point out that the nonlinearity $f$ is the almost ``optimal'' choice. First, for $\alpha\in({4}/{3},2)$, applying the generalized version of Lions-type theorem and the Nehari manifold, we show the existence of nonnegative Nehari-type ground sate solution for above equation. Second, for $\alpha\in(0,1)$, using the generalized version of Lions-type theorem and the Poho\v{z}aev manifold, we establish the existence of nonnegative Poho\v{z}aev-type ground state solution for above equation. Based on our new generalized version of Lions-type theorem, our works extend the results in Li-Su [Z. Angew. Math. Phys. {\bf 66} (2015)].