Existence and symmetry results for competing variational systems

TL;DR

This paper establishes the existence of ground-state solutions for a class of gradient systems with potential functions, and demonstrates symmetry properties of solutions in radially symmetric domains for the case of two equations.

Contribution

It proves existence results for ground states in general domains and symmetry properties for two-equation systems under radial symmetry assumptions.

Findings

Existence of ground-state solutions under suitable conditions.

Foliated Schwarz symmetry of solutions in radially symmetric domains for two equations.

Provides examples illustrating the abstract framework.

Abstract

In this paper we consider a class of gradient systems of type $$ -c_i \Delta u_i + V_i(x)u_i=P_{u_i}(u),\quad u_1,..., u_k>0 \text{in}\Omega, \qquad u_1=...=u_k=0 \text{on} \partial \Omega, $$ in a bounded domain $\Omega\subseteq \R^N$. Under suitable assumptions on $V_i$ and $P$, we prove the existence of ground-state solutions for this problem. Moreover, for $k=2$, assuming that the domain $\Omega$ and the potentials $V_i$ are radially symmetric, we prove that the ground state solutions are foliated Schwarz symmetric with respect to antipodal points. We provide several examples for our abstract framework.