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2019 On finding the ground state solution to the linearly coupled Brezis-Nirenberg system in high dimensions: the cooperative case
Yuanze Wu
Topol. Methods Nonlinear Anal. 53(2): 697-729 (2019). DOI: 10.12775/TMNA.2019.018

Abstract

Consider the following elliptic system \begin{equation*} \begin{cases} -\Delta u_i+\mu_i u_i=|u_i|^{2^*-2}u_i+\lambda \sum\limits_{j=1,j\not=i}^ku_j &\text{in }\Omega,\\ u_i=0,\quad i=1,\dots,k,&\text{on }\partial\Omega, \end{cases} \end{equation*} where $k\geq2$, $\Omega\subset\mathbb R^N$ ($N\geq4$) is a bounded domain with smooth boundary $\partial\Omega$, $2^*={2N}/({N-2})$ is the Sobolev critical exponent, $\mu_i\in\mathbb R$ for all $i=1,\dots,k$ are constants and $\lambda\in\mathbb R$ is a parameter. By the variational method, we mainly prove that the above system has a ground state for all $\lambda> 0$. Our results reveal some new properties of the above system that imply that the parameter $\lambda$ plays the same role as in the following well-known Brezis-Nirenberg equation \begin{equation*} \begin{cases} -\Delta u =\lambda u+ |u|^{2^*-2}u &\text{in }\Omega,\\ u=0 &\text{on }\partial\Omega, \end{cases} \end{equation*} and this system has a very similar structure of solutions as the above Brezis-Nirenberg equation for $\lambda$.

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Yuanze Wu. "On finding the ground state solution to the linearly coupled Brezis-Nirenberg system in high dimensions: the cooperative case." Topol. Methods Nonlinear Anal. 53 (2) 697 - 729, 2019. https://doi.org/10.12775/TMNA.2019.018

Information

Published: 2019
First available in Project Euclid: 11 May 2019

zbMATH: 07130716
MathSciNet: MR3983991
Digital Object Identifier: 10.12775/TMNA.2019.018

Rights: Copyright © 2019 Juliusz P. Schauder Centre for Nonlinear Studies

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Vol.53 • No. 2 • 2019
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