Segregated solutions for nonlinear Schr\"odinger systems with sublinear coupling terms

TL;DR

This paper proves the existence of infinitely many segregated solutions for a nonlinear Schrödinger system with sublinear coupling, introducing a new reduction method to handle nonsmooth nonlinearities and revealing a novel dead core phenomenon.

Contribution

It develops an advanced Lyapunov-Schmidt reduction framework to address sublinear couplings in Schrödinger systems, overcoming classical limitations and uncovering solution segregation phenomena.

Findings

Existence of infinitely many segregated solutions.

Introduction of a new reduction framework for nonsmooth nonlinearities.

Discovery of a dead core phenomenon with solutions vanishing in certain regions.

Abstract

We establish the existence of infinitely many nonnegative, segregated solutions for the sublinearly coupled Schr\"odinger system \begin{equation*} \left\{\begin{aligned}-\Delta u+K_1(x)u&=\mu u^{p-1}+ (\sigma_1+1)\beta u^{\sigma_1}v^{\sigma_2+1}, &x\in\mathbb{R}^N&, -\Delta v+K_2(x)v&=\nu v^{p-1}+(\sigma_2+1)\beta u^{\sigma_1+1}v^{\sigma_2}, &x\in\mathbb{R}^N&,\end{aligned}\right. \end{equation*}where $N \geq 2$, $p \in (2,2^*)$, $2^* = 2N/(N-2)$ ($2^* = \infty$ if $N=2$), $K_j$ are radial potentials, $\mu, \nu > 0$, $\beta \in \mathbb{R}$, and critically $\sigma_j \in (0,1)$. The sublinear coupling exponents $\sigma_j$ introduce fundamental challenges due to nonsmooth nonlinearities and singularities in standard reduction methods. To overcome this, we develop an enhanced Lyapunov-Schmidt reduction framework. By recasting the problem within a specially constructed metric space of local minimizers for an outer boundary value problem, we derive sharp a priori estimates enabling contraction mapping arguments. This approach circumvents the limitations of classical methods for sublinear couplings. We further uncover a novel "dead core" phenomenon: solutions $(u_\ell, v_\ell)$ exhibit non-strict positivity with topological segregation. Specially, for $N=2$ and large integers $\ell$, there exist radii $0 < R_1 < R_2$ such that $\text{supp } u_\ell \subset B_{R_2}(0)$, $\text{supp } v_\ell \subset \mathbb{R}^N \setminus B_{R_1}(0)$, and $u_\ell + v_\ell \to 0$ uniformly in $B_{R_2}(0) \setminus B_{R_1}(0)$ as $\ell \to \infty$. Our methodology provides a versatile framework for handling nonsmooth nonlinearities in reduction techniques.