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

TL;DR

This paper investigates energy bounds and existence of fully nontrivial solutions for a class of coupled semilinear elliptic systems with mixed couplings, including singularly perturbed problems with complex asymptotic behaviors.

Contribution

It provides simple conditions on the coupling matrix ensuring fully nontrivial solutions and establishes existence results with prescribed sign-changing components.

Findings

Existence of fully nontrivial solutions under certain matrix conditions.

Upper bounds for energy in systems with at most two blocks.

Solutions with different asymptotic behaviors in singular perturbation problems.

Abstract

We study the system of semilinear elliptic equations $$-\Delta u_i+ u_i = \sum_{j=1}^\ell \beta_{ij}|u_j|^p|u_i|^{p-2}u_i, \qquad u_i\in H^1(\mathbb{R}^N),\qquad i=1,\ldots,\ell,$$ where $N\geq 4$, $1<p<\frac{N}{N-2}$, 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 $$-\varepsilon^2\Delta u_i+ u_i = \sum_{j=1}^\ell \beta_{ij}|u_j|^p|u_i|^{p-2}u_i, \qquad u_i\in H^1_0(B_1(0)),\qquad i=1,\ldots,\ell,$$ in the unit ball, exhibiting two different kinds of asymptotic behavior: solutions whose components decouple as $\varepsilon\to 0$, and solutions whose components remain coupled all the way up to their limit.