Multiple solutions to weakly coupled supercritical elliptic systems

TL;DR

This paper proves the existence of infinitely many symmetric solutions to a class of supercritical elliptic systems, extending results to Hénon-type equations with power nonlinearities and symmetry considerations.

Contribution

It introduces new methods to establish multiple solutions for supercritical elliptic systems with symmetry and weighted nonlinearities, broadening understanding of such equations.

Findings

Existence of infinitely many symmetric solutions for supercritical systems.

Application to Hénon-type equations with new multiplicity results.

Solutions depend on symmetry properties and domain geometry.

Abstract

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 $\Omega$ is a bounded smooth domain in $\mathbb{R}^{N}$, $N\geq 3$, $\gamma\geq 0$, $\mu_{1},\mu_{2}>0$, $\lambda\in\mathbb{R}$, $\alpha, \beta>1$, $\alpha+\beta = p$, and $p\geq 2^{*}:=\frac{2N}{N-2}$. 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 $p\geq 2^*$ 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 H\'enon-type equation $$-\Delta w = |x_2|^\gamma \,|w|^{p-2}w \quad\text{in }\Omega, \qquad w=0 \quad\text{on }\partial\Omega.$$