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Spiked solutions for Schrödinger systems with Sobolev critical exponent: the cases of competitive and weakly cooperative interactions

In this paper we deal with the nonlinear Schrödinger system \[ -Δu_i =μ_i u_i^3 + βu_i \sum_{j\neq i} u_j^2 + λ_i u_i, \qquad u_1,\ldots, u_m\in H^1_0(Ω) \] in dimension 4, a problem with critical Sobolev exponent. In the competitive case ($β<0$ fixed or $β\to -\infty$) or in the weakly cooperative case ($β\geq 0$ small), we construct, under suitable assumptions on the Robin function associated to the domain $Ω$, families of positive solutions which blowup and concentrate at different points as $λ_1,\ldots, λ_m\to 0$. This problem can be seen as a generalization for systems of a Brezis-Nirenberg type problem.