Existence and nonexistence of entire solutions for non-cooperative cubic elliptic systems

Abstract

In this paper we deal with the cubic Schrödinger system $-\Delta u_i={\sum }_{j=1}^n{\beta }_{ij}{u}_j^2{u}_i$, $u_1,\dots ,u_n\ge 0$ in ${\mathbb{R}}^N(N\le 3)$, where $\beta =({\beta }_{i,j}{)}_{ij}$ is a symmetric matrix with real coefficients and ${\beta }_{ii}\ge 0$ for every $i=1,\dots ,n$. We analyse the existence and nonexistence of nontrivial solutions in connection with the properties of the matrix $\beta$, and provide a complete characterization in dimensions $N=1,2$. Extensions to more general power-type nonlinearities are given.

22 pages

References in corpus (1)

• Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition