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Ground state solution of critical Schrödinger equation with singular potential

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Y. Su is supported by the Key Program of University Natural Science Research Fund of Anhui Province (KJ2020A0292)
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  • In this paper, we consider the following Schrödinger equation with singular potential:

    $ \begin{equation*} \begin{aligned} -\Delta u + V(|x|)u = f(u),\ \, x\in \mathbb{R}^{N}, \end{aligned} \end{equation*} $

    where $ N\geqslant 3 $, $ V $ is a singular potential with parameter $ \alpha\in(0,2)\cup(2,\infty) $, the nonlinearity $ f $ involving critical exponent. First, by using the refined Sobolev inequality, we establish a Lions-type theorem. Second, applying Lions-type theorem and variational methods, we show the existence of ground state solution for above equation. Our result partially extends the results in Badiale-Rolando [Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 17 (2006)], and Su-Wang-Willem [Commun. Contemp. Math. 9 (2007)].

    Mathematics Subject Classification: Primary: 35J60; Secondary: 35J20.


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