Multiple solutions to a weakly coupled purely critical elliptic system in bounded domains

• Published in: arXiv.org
• Main Authors: Clapp, Mónica, Faya, Jorge
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 25.05.2018
• Subjects: Domains; Phase separation; Symmetry
• ISSN: 2331-8422

We study the weakly coupled critical elliptic system \begin{equation*} \begin{cases} -\Delta u=\mu_{1}|u|^{2^{*}-2}u+\lambda\alpha |u|^{\alpha-2}|v|^{\beta}u & \text{in }\Omega,\\ -\Delta v=\mu_{2}|v|^{2^{*}-2}v+\lambda\beta |u|^{\alpha}|v|^{\beta-2}v & \text{in }\Omega,\\ u=v=0 & \text{on }\partial\Omega, \end{cases} \end{equation*} where \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^{N}\), \(N\geq 3\), \(2^{*}:=\frac{2N}{N-2}\) is the critical Sobolev exponent, \(\mu_{1},\mu_{2}>0\), \(\alpha, \beta>1\), \(\alpha+\beta =2^{*}\) and \(\lambda\in\mathbb{R}\). We establish the existence of a prescribed number of fully nontrivial solutions to this system under suitable symmetry assumptions on \(\Omega\), which allow domains with finite symmetries, and we show that the positive least energy symmetric solution exhibits phase separation as \(\lambda\to -\infty\). We also obtain existence of infinitely many solutions to this system in \(\Omega=\mathbb{R}^N\).