Fully nontrivial solutions to elliptic systems with mixed couplings

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 $p\in(1,2).$