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

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 $$-\Delta u+u=\mu_i|u|^{p-2}u,\qquad u \in H^1(\mathbb{R}^N), \qquad u\neq 0.$$