Liouville-type theorems for elliptic Schrödinger systems associated with copositive matrices

Abstract

We study non-cooperative, multi-component elliptic Schrödinger systems arising in nonlinear optics and Bose-Einstein condensation phenomena.We here reconsider the more delicate case of systems of $m ≥ 3$ components.We prove a Liouville-type nonexistence theorem in space dimensions $n ≥3$ for homogeneous nonlinearities of degree $ p < n/(n-2)$, under the optimal assumption that the associated matrix is strictly copositive.This extends recent work of Tavares, Terracini, Verzini and Weth [22] and of Quittner and the author [17], where the results were limited to $n ≤ 2$ dimensions or to $m=2$ components. The proof of the Liouville theorem is done by combining and improving different arguments from [22] and [17], namelya feedback procedure based on Rellich-Pohozaev type identities and functional analytic inequalities on $S^{n-1}$, and suitable test-function arguments.We also consider a more general class of systems with gradient structure, for which our arguments show the triviality of solutions satisfying a suitable integral bound, a result which may be of independent interest.