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

TL;DR

This paper constructs solutions for a critical Sobolev exponent Schrödinger system in 4D, showing solutions concentrate at different points as parameters tend to zero, generalizing Brezis-Nirenberg problems to systems.

Contribution

It introduces a method to construct blow-up solutions for Schrödinger systems with critical exponent under competitive and weakly cooperative interactions.

Findings

Solutions blow up and concentrate at different points as parameters approach zero.

Construction of positive solutions under assumptions on the domain's Robin function.

Extension of Brezis-Nirenberg type results to coupled Schrödinger systems.

Abstract

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.