Segregated solutions for nonlinear Schrödinger systems with sublinear coupling terms

Abstract

We establish the existence of infinitely many nonnegative, segregated solutions for the sublinearly coupled Schrödinger system \begin{equation*} \left\{\begin{aligned}-Δu+K_1(x)u&=μu^{p-1}+ (σ_1+1)βu^{σ_1}v^{σ_2+1}, &x\in\mathbb{R}^N&, -Δv+K_2(x)v&=νv^{p-1}+(σ_2+1)βu^{σ_1+1}v^{σ_2}, &x\in\mathbb{R}^N&,\end{aligned}\right. \end{equation*}where $N\ge 2$, $p\in (2,2^{\ast })$, $2^{\ast }=2N/(N-2)$ ($2^{\ast }=\infty$ if $N=2$), $K_j$ are radial potentials, $\hat{I}¼,\hat{I}½>0$, ${\hat{I}}^2\in \mathbb{R}$, and critically $\ddot{I}{ƒ}_j\in (0,1)$. The sublinear coupling exponents $\ddot{I}{ƒ}_j$ 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.