Multiplicity of solutions for homogeneous elliptic systems with critical growth

TL;DR

This paper investigates the existence and multiplicity of nonnegative solutions for a class of elliptic systems with critical Sobolev growth, using variational methods and topological tools.

Contribution

It establishes new results on the number of solutions for elliptic systems with critical growth, employing variational techniques and Ljusternik-Schnirelmann theory.

Findings

Multiple nonnegative solutions are proven to exist.

The solutions are characterized using variational methods.

Topological methods are used to estimate the number of solutions.

Abstract

In this paper we are concerned with the number of nonnegative solutions of the elliptic system $$ {array}{ll} -\Delta u = Q_u(u,v) + 1/2{2^*} H_u(u,v),& {in} \Omega,\vdois\ -\Delta v = Q_v(u,v) + 1/{2^*} H_v(u,v),& {in} \Omega,\vdois\ u=v=0,& {on} \partial\Omega. \leqno{(P)} $$ where $\Omega \subset \mathbb{R}^N$ is a bounded smooth domain, $N \geq 3$, $2^*:=2N/(N-2)$ and $Q_u, H_u$ and $Q_v$, $H_v$ are the partial derivatives of the homogeneous functions $Q,\,H \in C^1(\mathbb{R}^2_+,\mathbb{R})$, where $ \mathbb{R}^2_+ := [0,\infty) \times [0,\infty)$. In the proofs we apply variational methods and Ljusternik-Schnirelmann theory.