Energy Estimates for Seminodal Solutions to an Elliptic System with Mixed Couplings

We study the system of semilinear elliptic equations

\begin{equation} \label{EqLim}    
\begin{array}{ll}
-\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,\end{array}
\end{equation}

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
\(-\epsilon^2\Delta u_i+ u_i = \displaystyle \sum_{j=1}^\ell \beta_{ij}|u_j|^p|u_i|^{p-2}u_i, \qquad u_i\in H^1_0(B_1(0)),\qquad i=1,\ldots,\ell,\)
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.

Los datos para la reunión de Zoom son los siguientes:

https://cuaieed-unam.zoom.us/j/89946525336
Meeting ID: 899 4652 5336