Fully nontrivial solutions to elliptic systems with mixed couplings

Abstract

We study the existence of fully nontrivial solutions to the system $$-\Delta u_i+{\lambda }_i{u}_i=\sum _{j=1}^{\ell }{\beta }_{ij}|u_j{|}^p|u_i{|}^{p-2}{u}_{i}in\Omega ,i=1,\dots ,\ell ,$$ 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 $i=2,\dots ,\ell$) in the purely cooperative case whenever $p\in (1,2).$