We consider the following nonlinear Schrödinger equation involving supercritical growth: {-Δ⁢u+V⁢(y)⁢u=Q⁢(y)⁢u2*-1+εin ⁢ℝN,u>0,u∈H1⁢(ℝN),\left\{\begin{aligned} &\displaystyle{-}\Delta u+V(y)u=Q(y)u^{2^{*}-1+% \varepsilon}&&\displaystyle\phantom{}\text{in }\mathbb{R}^{N},\\ &\displaystyle u>0,\quad u\in H^{1}(\mathbb{R}^{N}),\end{aligned}\right.{} where 2*=2⁢NN-2{2^{*}=\frac{2N}{N-2}} is the critical Sobolev exponent, N≥5{N\geq 5}, and V⁢(y){V(y)} and Q⁢(y){Q(y)} are bounded nonnegative functions in ℝN{\mathbb{R}^{N}}. By using the finite reduction argument and local Pohozaev-type identities, under some suitable assumptions on the functions V and Q , we prove that for ε>0{\varepsilon>0} is small enough, problem (*){(*)} has large number of bubble solutions whose functional energy is in the order ε-N-4(N-2)2.{\varepsilon^{-\frac{N-4}{(N-2)^{2}}}.}

中文翻译：

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具有超临界增长的Schrödinger方程的大能量气泡解

我们考虑以下涉及超临界增长的非线性Schrödinger方程：{-Δ⁢u+V⁢（y）⁢u=Q⁢（y）⁢u2* -1 +εin⁢ℝN，u> 0，u∈H1⁢（ℝN ），\ left \ {\开始{aligned}＆\ displaystyle {-} \ Delta u + V（y）u = Q（y）u ^ {2 ^ {*}-1 +％\ varepsilon} && \ displaystyle \ phantom {} \ text {in} \ mathbb {R} ^ {N}，\\＆\ displaystyle u> 0，\ quad u \ in H ^ {1}（\ mathbb {R} ^ {N}），\ end {aligned} \ right。{}。其中2 * =2⁢NN-2{2 ^ {*} = \ frac {2N} {N-2}}是关键的Sobolev指数，N≥5{N \ geq 5 }，而V⁢（y）{V（y）}和Q⁢（y）{Q（y）}是ℝN{\ mathbb {R} ^ {N}}中的有界非负函数。通过使用有限约简参数和局部Pohozaev型恒等式，在函数V和Q的一些适当假设下，我们证明对于ε> 0 {\ varepsilon> 0}足够小，问题（*）{（*）}具有大量的气泡溶液，其功能能为ε-N-4（N-2）2。