Multiple solutions to weakly coupled supercritical elliptic systems

Abstract

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 Henon-type equation $$\begin{aligned} -\Delta w = |x_2|^\gamma \,|w|^{p-2}w \quad \text {in }\Omega , \qquad w=0 \quad \text {on }\partial \Omega . \end{aligned}$$