Energy estimates for seminodal solutions to an elliptic system with mixed couplings

Abstract

We study the system of semilinear elliptic equations $$-\Delta u_i+u_i=\sum _{j=1}^{\ell }{\beta }_{ij}|u_j{|}^p|u_i{|}^{p-2}{u}_i,u_i\in H^1({\mathbb{R}}^N),i=1,\dots ,\ell ,$$ where $N\ge 4$, $1<p<\frac{N}{N-2}$, and the matrix $({\beta }_{ij})$ is symmetric and admits a block decomposition such that the entries within each block are positive or zero and all other entries are negative. We provide simple conditions on $({\beta }_{ij})$, which guarantee the existence of fully nontrivial solutions, i.e., solutions all of whose components are nontrivial. We establish existence of fully nontrivial solutions to the system having a prescribed combination of positive and nonradial sign-changing components, and we give an upper bound for their energy when the system has at most two blocks. We derive the existence of solutions with positive and nonradial sign-changing components to the system of singularly perturbed elliptic equations $$-{\epsilon }^2\Delta u_i+u_i=\sum _{j=1}^{\ell }{\beta }_{ij}|u_j{|}^p|u_i{|}^{p-2}{u}_i,u_i\in H_0^1(B_1(0)),i=1,\dots ,\ell ,$$ in the unit ball, exhibiting two different kinds of asymptotic behavior: solutions whose components decouple as $\epsilon \to 0$, and solutions whose components remain coupled all the way up to their limit.