Multiple solutions to weakly coupled supercritical elliptic systems

We study a weakly coupled supercritical elliptic system of the form \begin{equation*} \begin{cases} -\Delta u = |x_2|^\gamma \left(\mu_{1}|u|^{p-2}u+\lambda\alpha |u|^{\alpha-2}|v|^{\beta}u \right) & $\text{in }$\Omega,\ -\Delta v = |x_2|^\gamma \left(\mu_{2}|v|^{p-2}v+\lambda\beta |u|^{\alpha}|v|^{\beta-2}v \right) & $\text{in }$\Omega,\ u=v=0 & $\text{on }$\partial\Omega, \end{cases} \end{equation*} where $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$ 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 $x\in \mathrm{\Omega }$. Then, under some assumptions on $\mathrm{\Omega }$ 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 H'enon-type equation $$-\mathrm{\Delta }w=\mathrm{\mid }{x}_2{\mathrm{\mid }}^{\gamma }\mathrm{\mid }w{\mathrm{\mid }}^{p-2}w\text{in }\mathrm{\Omega },w=0\text{on }\mathrm{\partial }\mathrm{\Omega }\mathrm{.}$$