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

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

2010-07-18

arxiv.org/abs/1007.3007v1

Mathematics

Analysis of PDEs

22 pages

Scientific paper

In this paper we deal with the cubic Schr\"odinger system $ -\Delta u_i = \sum_{j=1}^n \beta_{ij}u_j^2 u_i$, $u_1,\dots,u_n \geq 0$ in $\mathbb{R}^N (N\leq 3)$, where $\beta=(\beta_{i,j})_{ij}$ is a symmetric matrix with real coefficients and $\beta_{ii}\geq 0$ for every $i=1,\ldots,n$. 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.

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