Coupled and uncoupled sign-changing spikes of singularly perturbed elliptic systems

TL;DR

This paper investigates the existence and asymptotic behavior of sign-changing solutions to a singularly perturbed elliptic system, revealing two distinct limit profiles as the perturbation parameter approaches zero.

Contribution

It introduces new solutions with prescribed positive and sign-changing components, demonstrating their asymptotic limits in the context of coupled elliptic systems.

Findings

Existence of solutions with mixed sign components.

Identification of two types of asymptotic profiles as perturbation vanishes.

Solutions converge to either a rescaled coupled system or an uncoupled system.

Abstract

We study the existence and asymptotic behavior of solutions having positive and sign-changing components to the singularly perturbed system of elliptic equations \begin{equation*} \begin{cases} -\varepsilon^2\Delta u_i+u_i=\mu_i|u_i|^{p-2}u_i + \sum\limits_{\substack{j=1 \\ j \not=i}}^\ell\lambda_{ij}\beta_{ij}|u_j|^{\alpha_{ij}}|u_i|^{\beta_{ij} -2}u_i,\\ u_i \in H^1_0(\Omega), \quad u_i\neq 0, \qquad i=1,\ldots,\ell, \end{cases} \end{equation*} in a bounded domain $\Omega$ in $\mathbb{R}^N$, with $N\geq 4$, $\varepsilon>0$, $\mu_i>0$, $\lambda_{ij}=\lambda_{ji}<0$, $\alpha_{ij}, \beta_{ij}>1$, $\alpha_{ij}=\beta_{ji}$, $\alpha_{ij} + \beta_{ij} = p\in (2,2^*)$, and $2^{*}:=\frac{2N}{N-2}$. If $\Omega$ is the unit ball we obtain solutions with a prescribed combination of positive and nonradial sign-changing components exhibiting two different types of asymptotic behavior as $\varepsilon\to 0$: solutions whose limit profile is a rescaling of a solution with positive and nonradial sign-changing components of the limit system \begin{equation*} \begin{cases} -\Delta u_i+u_i=\mu_i|u_i|^{p-2}u_i + \sum\limits_{\substack{j=1 \\ j \not=i}}^\ell\lambda_{ij}\beta_{ij}|u_j|^{\alpha_{ij}}|u_i|^{\beta_{ij} -2}u_i,\\ u_i \in H^1(\mathbb{R}^N), \quad u_i\neq 0, \qquad i=1,\ldots,\ell, \end{cases} \end{equation*} and solutions whose limit profile is a solution of the uncoupled system, i.e., after rescaling and translation, the limit profile of the $i$-th component is a positive or a nonradial sign-changing solution to the equation $$-\Delta u+u=\mu_i|u|^{p-2}u,\qquad u \in H^1(\mathbb{R}^N), \qquad u\neq 0.$$