Entire solutions to a strongly competitive nonlinear Schrödinger system
Pierpaolo Esposito, Pablo Figueroa, Angela Pistoia, Giusi Vaira
TL;DR
The authors address the existence of non-radial positive solutions for a strongly competitive Schrödinger system in $\mathbb{R}^N$ in the limit $\Lambda\to+\infty$. They build an explicit multi-peak approximate solution from translates of the scalar bubble $U$, arranged on two concentric regular $k$-gons and connecting rays, and perform a Lyapunov–Schmidt reduction combined with precise inner/outer interaction expansions. Two balancing equations fix peak spacings and scaling with $\Lambda$, enabling a finite-dimensional reduction in terms of translation parameters. For infinitely many admissible triplets $(m,n,k)$, the reduced system is solvable for large $\Lambda$, yielding non-radial positive solutions branching off from the approximate state $U^*$, thereby providing the first examples of non-synchronized, non-radial solutions in the whole space for strongly competitive systems.
Abstract
We build infinitely-many non-radial positive solutions to the Schrödinger system \begin{equation*} \left\{\begin{aligned} &-Δu_1+u_1=u_1^{{\mathfrak p} }-Λu_1^{a_1} u_2^{a_2}\ \hbox{in}\ \mathbb R^N\\ &-Δu_2+u_2=u_2^{{\mathfrak p} }-Λu_1^{b_1}u_2^{b_2} \ \hbox{in}\ \mathbb R^N\\ \end{aligned}\right. \end{equation*} with sub-critical $\mathfrak p$-growth as $Λ\to +\infty$. The profile of each component is the sum of several copies of the positive solution to $-ΔU+U=U^{{\mathfrak p} }$ in $\mathbb R^N$, centered at suitable {\em peaks} whose mutual distances diverge as $Λ$ increases. More precisely, given two concentric regular polygons with $k$ sides and very large radii, the peaks of the first component are arranged along the edges of the {\em outer} polygon, alternated with those of the second component, and along the $k$ rays joining the vertices of the two polygons. To the best of our knowledge, this provides the first example of non-radial positive solutions for strongly competitive Schrödinger systems in the whole space.