Spiked solutions for Schr\"odinger systems with Sobolev critical exponent: the cases of competitive and weakly cooperative interactions

In this paper we deal with the nonlinear Schr\"odinger system \[ -\Delta u_i =\mu_i u_i^3 + \beta u_i \sum_{j\neq i} u_j^2 + \lambda_i u_i, \qquad u_1,\ldots, u_m\in H^1_0(\Omega) \] in dimension 4, a problem with critical Sobolev exponent. In the competitive case ($\beta<0$ fixed or $\beta\to -\infty$) or in the weakly cooperative case ($\beta\geq 0$ small), we construct, under suitable assumptions on the Robin function associated to the domain $\Omega$, families of positive solutions which blowup and concentrate at different points as $\lambda_1,\ldots, \lambda_m\to 0$. This problem can be seen as a generalization for systems of a Brezis-Nirenberg type problem.