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 $Ω$ is a bounded smooth domain in $\mathbb{R}^{N}$, $N\geq 3$, $γ\geq 0$, $μ_{1},μ_{2}>0$, $λ\in\mathbb{R}$, $α, β>1$, $α+β= p$, and $p\geq 2^{*}:=\frac{2N}{N-2}$. We assume that $Ω$ 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 $x\inΩ$. Then, under some assumptions on $Ω$ and $F$, we establish the existence of infinitely many fully nontrivial $G$-invariant solutions to this system for $p\geq 2^*$ up to some value which depends on the symmetries and on $γ$. Our results apply, in particular, to the system with pure power nonlinearity ($γ=0$), and yield new existence and multiplicity results for the supercritical Hénon-type equation $$-Δw = |x_2|^γ\,|w|^{p-2}w \quad\text{in }Ω, \qquad w=0 \quad\text{on }\partialΩ.$$