Multiple solutions to weakly coupled supercritical elliptic systems

  • Omar Cabrera
  • Mónica ClappEmail author


We study a weakly coupled supercritical elliptic system of the form
$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u = |x_2|^\gamma \left( \mu _{1}|u|^{p-2}u+\lambda \alpha |u|^{\alpha -2}|v|^{\beta }u \right) &{}\quad \text {in }\Omega ,\\ -\Delta v = |x_2|^\gamma \left( \mu _{2}|v|^{p-2}v+\lambda \beta |u|^{\alpha }|v|^{\beta -2}v \right) &{}\quad \text {in }\Omega ,\\ u=v=0 &{}\quad \text {on }\partial \Omega , \end{array}\right. } \end{aligned}$$
where \(\Omega \) is a bounded smooth domain in \({\mathbb {R}}^{N}\), \(N\ge 3\), \(\gamma \ge 0\), \(\mu _{1},\mu _{2}>0\), \(\lambda \in {\mathbb {R}}\), \(\alpha , \beta >1\), \(\alpha +\beta = p\), and \(p\ge 2^{*}:=\frac{2N}{N-2}\). We assume that \(\Omega \) is invariant under the action of a group G of linear isometries, \({\mathbb {R}}^{N}\) is the sum \(F\oplus F^\perp \) of G-invariant linear subspaces, and \(x_2\) is the projection onto \(F^\perp \) of the point \(x\in \Omega \). Then, under some assumptions on \(\Omega \) and F, we establish the existence of infinitely many fully nontrivial G-invariant solutions to this system for \(p\ge 2^*\) up to some value which depends on the symmetries and on \(\gamma \). Our results apply, in particular, to the system with pure power nonlinearity (\(\gamma =0\)) and yield new existence and multiplicity results for the supercritical Hénon-type equation
$$\begin{aligned} -\Delta w = |x_2|^\gamma \,|w|^{p-2}w \quad \text {in }\Omega , \qquad w=0 \quad \text {on }\partial \Omega . \end{aligned}$$


Weakly coupled elliptic system Bounded domain Supercritical nonlinearity Hénon-type equation Phase separation 

Mathematics Subject Classification

35J47 35B33 35B40 35J50 



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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Instituto de MatemáticasUniversidad Nacional Autónoma de MéxicoCoyoacánMexico

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