Positive solution to Schrödinger equation with singular potential and double critical exponents

Abstract

In this paper, we consider the following Schrödinger equation with singular potential and double critical exponents: -\Delta u + \frac{A}{|x|^{\alpha}}u = |u|^{2^{*}-2}u + |u|^{p-2}u + \lambda |u|^{2_{\alpha}^{*}-2}u,\quad x\in \mathbb{R}^{N}, where N\geq 3 , \alpha\in(0,2) , p\in(\frac{2N-4+2\alpha}{N-2},2^{*}) , and A,\lambda>0 are two real constants, and 2^{*}=\frac{2N}{N-2} is the Sobolev critical exponent, and 2_{\alpha}^{*}=2+\frac{4\alpha}{2N-2-\alpha} is the critical exponent with respect to the parameter \alpha . First, using the refined Sobolev inequality, we establish the Lions type theorem. Second, we prove that any nonnegative weak solutions of above equation satisfy Pohozaev type identity. Finally, by using perturbation method, Pohozaev type identity and Lions type theorem, we show the existence of positive solution to above equation. We point out that the double critical exponents is an new phenomenon, and we are the first to consider it. Our result partial extends the results in Badiale and Rolando [Rend. Lincei Mat. Appl. 17 (2006)], and Su, Wang and Willem [Commun. Contemp. Math. 9 (2007)].