In this paper, we employ an enhanced version of the Lyapunov–Schmidt reduction method to study a particular class of nonlinear Schrödinger systems featuring sublinear coupling terms. Under suitable assumptions, we establish the existence of infinitely many nonnegative, segregated solutions for the system \(\begin{aligned} \left\{ \begin{aligned}-\Delta u+K_1(x)u&=\mu u^{p-1}+ (\sigma _1+1)\beta u^{\sigma _1}v^{\sigma _2+1},&x\in \mathbb {R}^N&, \\ -\Delta v+K_2(x)v&=\nu v^{p-1}+(\sigma _2+1)\beta u^{\sigma _1+1}v^{\sigma _2},&x\in \mathbb {R}^N&,\end{aligned}\right. \end{aligned}\) where \(N\ge 2\) , \( p\in (2,2^*) \) with \( 2^* = \frac{2N}{N-2} \) denoting the critical Sobolev exponent if \( N \ge 3 \) (and \( 2^* = \infty \) when \( N = 2 \) ). The functions \( K_j(x) \) , \( j = 1, 2 \) , are radially symmetric potential functions, the exponents \( \sigma _j \in (0,1) \) correspond to sublinear coupling terms, \( \mu > 0 \) and \( \nu > 0 \) are given constants, and \( \beta \in \mathbb {R} \) acts as the coupling coefficient.
The range of the exponents \(\sigma _j\) introduces substantial challenges to classical reduction methods, primarily due to the nonsmoothness and sublinearity inherent in the coupling terms. To address these difficulties, we introduce a novel approach that recasts the reduction process as a fixed point problem defined on an appropriately constructed metric space. This space is formed by local minimizers of an associated outer boundary value problem and is furnished with crucial a priori estimates, which together enable us to verify the contraction mapping property.
Moreover, we identify a novel phenomenon in the sublinearly coupled regime: the constructed solutions \((u_\ell , v_\ell )\) exhibit a distinct “dead core” behavior, characterized by non-strict positivity. In particular, for \(N = 2\) , we show that the supports of the components separate as follows: for each sufficiently large integer \(\ell \) , there exist radii \(0< R_1 < R_2\) , depending on \(\ell \) , 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 \(u_\ell + v_\ell \rightarrow 0\) uniformly in the annular region \(B_{R_2}(0) \setminus B_{R_1}(0)\) as \(\ell \rightarrow \infty \) .
We believe that the framework developed here has broad applicability and can be used to tackle other problems involving similar nonsmooth nonlinearities.
