# Least energy nodal solutions of Hamiltonian elliptic systems with Neumann boundary conditions

## TL;DR

This paper investigates the existence, symmetry, and qualitative properties of least energy nodal solutions to Hamiltonian elliptic systems with Neumann boundary conditions, introducing a novel transformation to handle non-standard settings.

## Contribution

It introduces a new $L^t$-norm-preserving transformation for analyzing solutions, revealing symmetry-breaking and monotonicity properties in Neumann problems.

## Key findings

- Least energy solutions are foliated Schwarz symmetric.
- Radial solutions are strictly monotone due to the transformation.
- The approach applies to scalar models, providing new insights.

## Abstract

We study existence, regularity, and qualitative properties of solutions to the system \[ -\Delta u = |v|^{q-1} v\quad \text{ in }\Omega,\qquad -\Delta v = |u|^{p-1} u\quad \text{ in }\Omega,\qquad \partial_\nu u=\partial_\nu v=0\quad \text{ on }\partial\Omega, \] with $\Omega\subset \mathbb R^N$ bounded; in this setting, all nontrivial solutions are sign changing. Our proofs use a variational formulation in dual spaces, considering sublinear $pq< 1$ and superlinear $pq>1$ problems in the subcritical regime. In balls and annuli we show that least energy solutions (l.e.s.) are foliated Schwarz symmetric and, due to a symmetry-breaking phenomenon, l.e.s. are not radial functions; a key element in the proof is a new $L^t$-norm-preserving transformation, which combines a suitable flipping with a decreasing rearrangement. This combination allows us to treat annular domains, sign-changing functions, and Neumann problems, which are non-standard settings to use rearrangements and symmetrizations. In particular, we show that our transformation diminishes the (dual) energy and, as a consequence, radial l.e.s. are strictly monotone. We also study unique continuation properties and simplicity of zeros. Our theorems also apply to the scalar associated model, where our approach provides new results as well as alternative proofs of known facts.