Least Energy Solutions for Cooperative and Competitive Schr\"odinger Systems with Neumann Boundary Conditions

We study the following gradient elliptic system with Neumann boundary conditions \begin{equation*} -\Delta u + \lambda_1 u = u^3 + \beta uv^2, \ -\Delta v + \lambda_2 v = v^3 + \beta u^2 v \ \text{in } \Omega,\qquad \frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = 0 \ \text{on } \partial \Omega, \end{equation*} where $\Omega \subset \mathbb{R}^N$ is a bounded $C^2$ domain with $N \leq 4$, 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 $N \leq 3$ 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.