Fully nontrivial solutions to elliptic systems with mixed couplings

TL;DR

This paper establishes existence and multiplicity of fully nontrivial solutions for elliptic systems with mixed attractive and repulsive interactions, covering subcritical, critical, and supercritical nonlinear growth in various domains.

Contribution

It introduces new results on solutions where all components are nontrivial, including synchronized solutions in cooperative cases for specific nonlinearities.

Findings

Existence of fully nontrivial solutions in various growth regimes.

Multiplicity results for solutions with complex interaction matrices.

Existence of synchronized solutions in cooperative systems for p in (1,2).

Abstract

We study the existence of fully nontrivial solutions to the system $$-\Delta u_i+ \lambda_iu_i = \sum\limits_{j=1}^\ell \beta_{ij}|u_j|^p|u_i|^{p-2}u_i\ \hbox{in}\ \Omega, \qquad i=1,\ldots,\ell,$$ in a bounded or unbounded domain $\Omega$ in $\mathbb R^N,$ $N\ge 3$. The $\lambda_i$'s are real numbers, and the nonlinear term may have subcritical ($1<p<\frac{N}{N-2}$), critical ($p=\frac{N}{N-2}$), or supercritical growth ($p>\frac{N}{N-2}$). The matrix $(\beta_{ij})$ is symmetric and admits a block decomposition such that the diagonal entries $\beta_{ii}$ are positive, the interaction forces within each block are attractive (i.e., all entries $\beta_{ij}$ in each block are non-negative) and the interaction forces between different blocks are repulsive (i.e., all other entries are non-positive). We obtain new existence and multiplicity results of fully nontrivial solutions, i.e., solutions where every component $u_i$ is nontrivial. We also find fully synchronized solutions (i.e., $u_i=c_i u_1$ for all $i=2,\ldots,\ell$) in the purely cooperative case whenever $p\in(1,2).$