Existence and symmetry of least energy nodal solutions for Hamiltonian elliptic systems

TL;DR

This paper proves the existence of least energy nodal solutions for Hamiltonian elliptic systems with weights, analyzes their symmetry properties, and demonstrates symmetry breaking in certain parameter regimes, advancing understanding of coupled elliptic systems.

Contribution

It establishes the existence of least energy nodal solutions with symmetry properties for weighted Hamiltonian elliptic systems, including symmetry breaking results, a novel contribution in this field.

Findings

Solutions exhibit foliated Schwarz symmetry in symmetric domains.

Symmetry breaking occurs when solutions behave like single equations.

Results are new even for unweighted Lane-Emden systems.

Abstract

In this paper we prove existence of least energy nodal solutions for the Hamiltonian elliptic system with H\'enon-type weights \[ -\Delta u = |x|^{\beta} |v|^{q-1}v, \quad -\Delta v =|x|^{\alpha}|u|^{p-1}u\quad { in } \Omega, \qquad u=v=0 { on } \partial \Omega, \] where $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$, $N\geq 1$, $\alpha, \beta \geq 0$ and the nonlinearities are superlinear and subcritical, namely \[ 1> \frac{1}{p+1}+\frac{1}{q+1}> \frac{N-2}{N}. \] When $\Omega$ is either a ball or an annulus centred at the origin and $N \geq 2$, we show that these solutions display the so-called foliated Schwarz symmetry. It is natural to conjecture that these solutions are not radially symmetric. We provide such a symmetry breaking in a range of parameters where the solutions of the system behave like the solutions of a single equation. Our results on the above system are new even in the case of the Lane-Emden system (i.e. without weights). As far as we know, this is the first paper that contains results about least energy nodal solutions for strongly coupled elliptic systems and their symmetry properties.