Multiple solutions to weakly coupled supercritical elliptic systems

Abstract

We study a weakly coupled supercritical elliptic system of the form \begin{equation*} \begin{cases} -Δu = |x_2|^γ\left(μ_{1}|u|^{p-2}u+λα|u|^{α-2}|v|^βu \right) & \text{in }Ω,\\ -Δv = |x_2|^γ\left(μ_{2}|v|^{p-2}v+λβ|u|^α|v|^{β-2}v \right) & \text{in }Ω,\\ u=v=0 & \text{on }\partialΩ, \end{cases} \end{equation*} 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 $p\ge 2^{\ast }:=\frac{2N}{N-2}$. 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 $p\ge 2^{\ast }$ 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 $$-\Delta w=|x_2{|}^{\gamma }|w{|}^{p-2}w\text{in }\Omega ,w=0\text{on }\partial \Omega .$$