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

Abstract

We study the system of semilinear elliptic equations

$$\begin{aligned} -\Delta u_i+ u_i = \displaystyle \sum _{j=1}^\ell \beta _{ij}|u_j|^p|u_i|^{p-2}u_i, \qquad u_i\in H^1(\mathbb {R}^N),\qquad i=1,\ldots ,\ell , \end{aligned}$$

$N\ge 4$, $1<p<\frac{N}{N-2}$, 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

$$\begin{aligned} -\varepsilon ^2\Delta u_i+ u_i = \displaystyle \sum _{j=1}^\ell \beta _{ij}|u_j|^p|u_i|^{p-2}u_i, \qquad u_i\in H^1_0(B_1(0)),\qquad i=1,\ldots ,\ell , \end{aligned}$$

in the unit ball, exhibiting two different kinds of asymptotic behavior: solutions whose components decouple as $\varepsilon \rightarrow 0$, and solutions whose components remain coupled all the way up to their limit.