Existence and symmetry results for competing variational systems

Abstract

In this paper we consider a class of gradient systems of type $$-c_i\Delta u_i+V_i(x)u_i=P_{u_i}(u),u_1,...,u_k>0\text{in}\Omega ,u_1=...=u_k=0\text{on}\partial \Omega ,$$ in a bounded domain $\Omega \subseteq {\mathbb{R}}^N$. 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.

21 pages, 0 figures

References in corpus (1)

• On the symmetry of minimizers