The main aim of this talk is to discuss the existence of nontrivial (and non semi-trivial) least energy solutions for a Neumann elliptic system with a critical nonlinearity, characterized by a cooperative-competitive behaviour, namely
\[ \begin{cases} -\Delta u+\lambda_1 u=u^3+β uv^2 & \text{ in } \Omega\\ -\Delta v+\lambda_2 v=v^3+\beta u^2v & \text{ in } \Omega\\ \frac{\partial u}{\partial\nu}=\frac{\partial v}{\partial \nu}=0 &\text{ on } \partial\Omega,\\ \end{cases} \label{eq:pbeta} \tag{$\mathcal P_\beta$} \] where $\Omega\subset \mathbb{R}^4$ is a $C^2$ bounded domain, and $\lambda_1,\lambda_2>0$ and the parameter $\beta\in\mathbb{R}$ captures the essence of cooperation-competition, assuming positive or negative values respectively.

The approach is variational and the idea is to minimize the energy functional on a suitable manifold of the Nehari type. In addition, to deal with the critical power, we estimate the energy level, using the solutions of $-\Delta w=w^3$ in $ \mathbb{R} ^4$ and the solution for the scalar equations $-\Delta u_i+\lambda_iu_i=u_i^3$ in $\Omega$, to establish a compactness condition based on the classical Cherrier's inequality: if $\partial\Omega\in C^1$ then for each $\varepsilon\gt 0$ there exists $M_{\varepsilon}>0$ such that $$\|u\|_{2^*}\le\left(\frac{2^{2/N}}{S}+\varepsilon\right)^{1/2}\|\nabla u\|_2+M_{\varepsilon}\|u\|_2,\ \ ∀u\in H^1(\Omega),$$ where $S$ is the best Sobolev constant. Additionally, I will discuss the more difficult cases in which $\lambda_1,\lambda_2\le0$, that I have started to study recently.