Least Energy Solutions for Cooperative and Competitive Schrödinger Systems with Neumann Boundary Conditions

We study the following gradient elliptic system with Neumann boundary conditions \begin{equation*} -Δu + λ_1 u = u^3 + βuv^2, \ -Δv + λ_2 v = v^3 + βu^2 v \ $\text{in }$ Ω,\qquad \frac{\partial u}{\partial ν} = \frac{\partial v}{\partial ν} = 0 \ $\text{on }$ \partial Ω, \end{equation*} where $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded $C^2$ domain with $N\le 4$, and $\nu$ denotes the outward unit normal on the boundary. We investigate the existence of non-constant least energy solutions in both the cooperative ($\beta >0$) and the competitive ($\beta <0$) regimes, considering both the definite and the indefinite case, namely ${\lambda }_1,{\lambda }_2\in \mathbb{R}$. We emphasize that our analysis includes both the subcritical case $N\le 3$ and the critical case $N=4$. Depending on the values of $\beta ,{\lambda }_1,{\lambda }_2$, the least energy solution is obtained either via a linking theorem, by minimizing over a suitable Nehari manifold, or by direct minimization on the set of all non-trivial weak solutions. Our results and techniques can be also adapted to cover some previously untreated cases for Dirichlet conditions.