Critical Kirchhoff-type equation with singular potential - Su, Yu; Liu, Senli (Topological Methods in Nonlinear Analysis, 2023)
In this paper, we deal with the following Kirchhoff-type equation:\begin{equation*}-\bigg(1+\int_{\mathbb{R}^{3}}| abla 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 nonnegativeNehari-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 groundstate solution for above equation. Based on our new generalized versionof Lions-type theorem, our works extend the results in Li-Su [Z. Angew. Math. Phys. {\bf 66} (2015)].