Ground States for a Nonlinear Schrödinger System with Sublinear Coupling Terms

Abstract

We study the existence of ground states for the coupled Schrödinger system(the so-called "symmetric attractive case") and 1 < q < n/(n − 2) + . We prove the existence of a nonnegative ground state (u * 1 , . . . , u * d ) with u * i radially decreasing. Moreover we show that, for 1 < q < 2, such ground states are positive in all dimensions and for all values of the parameters.

Ground-states for systems ofMcoupled semilinear Schrödinger equations with attraction–repulsion effects: Characterization and perturbation results

“…Note that the first part of corollary 2 is already known (see [12] and [10]). Also, in corollary 3, if ω 1 " ω M , the result is a particular case of [8] and [14]. Even so, we prove these results for two reasons: first, the proof is very simple when one looks from this pertubative perspective; second, the approach is rather different in nature and it deals only with continuity properties, which may have a greater capacity of generalization to other systems.…”

Existence of least energy positive solutions to Schrödinger systems with mixed competition and cooperation terms: the critical case

“…In [39,40], by dividing the d components into m groups, the authors proved that system (1.1) has a least energy positive solution under appropriate assumptions on β ij . Other related and recent results in the subcritical case can be found in [14,15,16,18,29,31,47] and references, where we stress that the first two mentioned papers deal with a general p.…”

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

“…One of the interesting features of these systems in the fact that they admit solutions with trivial components; for this reason, the systems are sometimes called weakly coupled. It should be noted that both the sublinear/superlinear character of the exponent p [19,24] or the number of the equations [12] influence the existence results of nontrivial least energy solutions. As for the critical case p = N/(N − 2), its study is recent, starting from the paper by Chen and Zou [10], for a system with m = 2 equations and exponent p = 2.…”