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)].

中文翻译：

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具有奇异势和双临界指数的Schrödinger方程的正解

在本文中，我们考虑以下具有奇异电位和双临界指数的Schrödinger方程：$$-\ Delta u + \ frac {A} {| x | ^ {\ alpha}} u = | u | ^ {2 ^ { *}-2} u + | u | ^ {p-2} u + \ lambda | u | ^ {2 _ {\ alpha} ^ {*}-2} u，\ quad x \ in \ mathbb {R} ^ {N}，$$其中$ N \ geq 3 $，$ \ alpha \ in（0,2）$，$ p \ in（\ frac {2N-4 + 2 \ alpha} {N-2}，2 ^ {*}）$和$ A，\ lambda> 0 $是两个实常数，而$ 2 ^ {*} = \ frac {2N} {N-2} $是Sobolev临界指数，而$ 2 _ {\ alpha } ^ {*} = 2+ \ frac {4 \ alpha} {2N-2- \ alpha} $是相对于参数$ \ alpha $的关键指数。首先，使用完善的Sobolev不等式，我们建立Lions型定理。其次，我们证明上述方程的任何非负弱解都满足Pohozaev类型恒等式。最后，通过摄动法，用Pohozaev型恒等式和Lions型定理，我们证明了上面方程的正解的存在。我们指出，双重临界指数是一个新现象，我们是第一个考虑它的人。我们的结果部分扩展了Badiale和Rolando [Rend。的结果。Lincei Mat。应用 第17页（2006年），以及Su，Wang和Willem [Commun。同时期 数学。9（2007）]。