Fully sign-changing Nehari constraint vs sign-changing solutions of a competitive Schrödinger system

where N ≥ 1, m ≥ 2. Nonlinear Schrödinger systems with competing interactions have been extensively studied in recent years, motivated both by physical models such as Bose-Einstein condensates and nonlinear optics, and by the rich mathematical structure of coupled elliptic equations. Among the various phenomena that may occur in this context, a particularly interesting situation arises when the nonlinear potentials are localized in small regions that shrink to isolated points as a small parameter ε > 0 tends to zero. In this case, solutions may concentrate around finitely many prescribed points and exhibit a delicate interaction between the different components.

We now state the structural assumptions that will be used throughout the paper. They are standard in this context and reflect the subcritical regime and the competitive nature of the system.

(A 1 ) 1 < p < 2 * 2 if N ≥ 3, and p > 1 if N = 1, 2, where 2 * = 2N N -2 if N ≥ 3.

(A 2 ) µ i > 0, i = 1, . . . , m, λ ij = λ ji < 0, for all i ̸ = j.

and there exist exactly pairwise distinct points y 1 , . . . , y m ∈ R N such that

for all i = 1, . . . , m.

In this setting, the solutions u i may be interpreted as standing wave profiles of different species which are attracted to regions where Q ε is positive and repelled from their complement. The assumption λ ij < 0 means that distinct components repel each other, which in turn favors spatial segregation, so the system (1.1) is competitive. Let ρ ∈ C ∞ 0 (R N ) be the standard compactly supported bump

and define ϕ(x) := ρ(x) ρ(0) .

Then ϕ ∈ C ∞ 0 (R N ), ϕ ≥ 0, supp(ϕ) ⊂ B 1 (0), ϕ(0) = 1, and ϕ(x) < 1 for all x ̸ = 0. For each i = 1, . . . , m define

and set

Then Q satisfies (A 3 ).

In the scalar case, a fundamental contribution is due to Ackermann and Szulkin [1], who showed that when the positive region of the nonlinear coefficient collapses to isolated points, every nontrivial solution concentrates at one of these cores, and ground states select a single core without splitting their mass. In this framework, the sign structure of the nonlinearity is the sole driver of localization, without any periodicity or symmetry assumption on the linear part. Since then, their approach has been extended in various directions, including problems on the whole space and coupled systems; see, e.g., [8,12,29] and the references therein. We point out that these works focus primarily on positive solutions or the least energy solutions, for which the Mountain Pass Theorem ( [28]) and the maximum principles are available. Very recently, in [12], Clapp, Saldaña and Szulkin studied the system (1.1) with a single or multiple shrinking domains, the existence of nonnegative least energy solution and concentration behavior were obtained as ε → 0. Furthermore, the authors characterized the limit profile and described how the components either decouple or remain coupled depending on the geometry of the attraction centers. Related results for scalar equations, including concentration of semiclassical states and the description of their limit profiles, were obtained in earlier works such as [1,15], and see [26] and the references therein for phase separation phenomena of competitive systems. We refer [10] to the existence of concentrating positive solutions via a Lyapunov Schmidt reduction strategy.

However, much less is known about sign-changing solutions for systems of the form (1.1). Even for the scalar equation, constructing nodal solutions requires a refined variational approach based on nodal Nehari sets and careful control of the positive and negative parts of the solutions. As far as we known, there are only two papers [8,10] concerned with this topic. In [8], Clapp, Hernández-Santamaría and Saldaña obtained the existence and concentration of nodal solutions via the nodal Nehari manifold method, and characterized the symmetries and the polynomial decay of the least-energy nodal limiting profiles. In [10], Clapp, Pistoia and Saldaña established the existence of concentrating nodal solutions via a Lyapunov Schmidt reduction method. But for systems, the situation is substantially more delicate, since one has to keep track simultaneously of the sign structure of each component and of the competitive couplings between different components. It seems that there is no work concerned with the existence and concentration of sign-changing solutions of the system (1.1).

First, we state the meaning of sign-changing solutions of (1.1).

Definition 1.1 A solution u = (u 1 , u 2 , • • • , u m ) of (1.1) is called sign-changing if for each i = 1, . . . , m, both the positive part u + i = max{u i , 0} and the negative part u - i = min{u i , 0} are nonzero in H 1 (R N ), i.e., ∥u ± i ∥ H 1 > 0.

The purpose of this paper is to develop a variational framework to seek for infinitely many sign-changing solutions of (1.1), and to analyze their concentration behavior