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 $$-\mathrm{\Delta }{u}_i={\mu }_i{u}_i^3+\beta u_i\sum _{j\mathrm{\ne }i}{u}_j^2+{\lambda }_i{u}_i,u_1,\dots ,u_m\in H_0^1(\mathrm{\Omega })$$ in dimension 4, a problem with critical Sobolev exponent. In the competitive case ($\beta <0$ fixed or $\beta\to -\infty$)orintheweaklycooperativecase($\beta\geq 0$ small), we construct, under suitable assumptions on the Robin function associated to the domain $\mathrm{\Omega }$, families of positive solutions which blowup and concentrate at different points as ${\lambda }_1,\dots ,{\lambda }_m\to 0$. This problem can be seen as a generalization for systems of a Brezis-Nirenberg type problem.