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2021 A singular perturbed problem with critical Sobolev exponent
Mengyao Chen, Qi Li
Topol. Methods Nonlinear Anal. 58(1): 181-207 (2021). DOI: 10.12775/TMNA.2020.067

Abstract

This paper deals with the following nonlinear elliptic problem \begin{equation}\tag{0.1} -\varepsilon^2\Delta u+\omega V(x)u=u^{p}+u^{2^{*}-1},\quad u> 0\quad\text{in}\ \mathbb R^N, \end{equation} where $\omega\in\mathbb R^{+}$, $N\geq 3$, $p\in (1,2^{*}-1)$ with $2^{*}={2N}/({N-2})$, $\varepsilon> 0$ is a small parameter and $V(x)$ is a given function. Under suitable assumptions, we prove that problem (0.1) has multi-peak solutions by the Lyapunov-Schmidt reduction method for sufficiently small $\varepsilon$, which concentrate at local minimum points of potential function $V(x)$. Moreover, we show the local uniqueness of positive multi-peak solutions by using the local Pohozaev identity.

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Mengyao Chen. Qi Li. "A singular perturbed problem with critical Sobolev exponent." Topol. Methods Nonlinear Anal. 58 (1) 181 - 207, 2021. https://doi.org/10.12775/TMNA.2020.067

Information

Published: 2021
First available in Project Euclid: 21 September 2021

Digital Object Identifier: 10.12775/TMNA.2020.067

Rights: Copyright © 2021 Juliusz P. Schauder Centre for Nonlinear Studies

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Vol.58 • No. 1 • 2021
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