Sign-changing solutions of competition-diffusion elliptic systems and optimal partition problems

Abstract

In this paper we prove the existence of infinitely many sign-changing solutions for the system of $m$ Schrödinger equations with competition interactions $$-\Delta u_i+a_i{u}_i^3+\beta u_i\sum _{j\ne i}{u}_j^2={\lambda }_{i,\beta }{u}_i{u}_i\in H_0^1(\Omega ),i=1,...,m$$ where $\Omega$ is a bounded domain, $\beta >0$ and $a_i\ge 0\forall i.$ Moreover, for $a_i=0$, we show a relation between critical energies associated with this system and the optimal partition problem $${\underset{{\omega }_i\subset \Omega \text{open}}{\inf }}_{{\omega }_i\cap {\omega }_j=\varnothing \forall i\ne j}\sum _{i=1}^m{\lambda }_{k_i}({\omega }_i),$$ where ${\lambda }_{k_i}(\omega )$ denotes the $k_i$--th eigenvalue of $-\Delta$ in $H_0^1(\omega )$. In the case $k_i\le 2$ we show that the optimal partition problem appears as a limiting critical value, as the competition parameter $\beta$ diverges to $+\infty$.

25 pages

References in corpus (2)

• Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition
• Regularity of the nodal set of segregated critical configurations under a weak reflection law

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