From subcritical to critical coupling

Coupled nonlinear Schrödinger equations arise naturally in several physical models, particularly in nonlinear optics. From a mathematical perspective, they also provide a rich setting in which the interaction between different components can strongly influence the existence and multiplicity of solutions.

Our paper, Schrödinger systems with critical coupling in Rᴺ, grew naturally from our previous work on weakly coupled Schrödinger systems with subcritical interaction. The question we wanted to address was what happens when the coupling reaches the critical Sobolev growth.

More precisely, the coupling exponents satisfy

α + μ = 2^* = 2N/(N − 2).

This apparently simple change has an important consequence: the compactness available in the subcritical setting is lost. As a result, standard variational arguments cannot be applied directly.

The main difficulty: loss of compactness

Our approach is variational and is based on the energy functional restricted to the Nehari manifold.

One of the central steps is the analysis of Palais–Smale sequences. Because the coupling has critical growth, weak convergence does not automatically provide the strong convergence needed to obtain critical points of the functional.

To deal with this difficulty, we establish a Brezis–Lieb type decomposition adapted to the mixed critical term. This makes it possible to separate the weak limit from the remaining part of a Palais–Smale sequence and to identify an energy threshold below which compactness can be recovered.

The threshold is determined by a Sobolev-type constant associated with the critical interaction between the two components.

Keeping the variational levels below the critical threshold

Recovering compactness below a certain energy level is only part of the problem. We also need to prove that the variational levels associated with the solutions actually remain below this threshold.

For this purpose, we use suitable test functions based on Sobolev extremals and derive estimates inspired by the classical Brezis–Nirenberg argument.

These estimates are a key ingredient in overcoming the difficulties created by the critical coupling.

The strength and nature of the coupling matter

The coupling parameter β > 0 leads to different variational regimes.

For sufficiently small β, we obtain a positive solution through a mountain-pass construction on the Nehari manifold. The corresponding minimax path connects the two semitrivial solutions of the system.

For sufficiently large β, under the additional assumption required in dimension three, we prove the existence of a positive least-energy solution.

The nature of the coupling with respect to each component is also important. When the coupling is sublinear with respect to one of the variables, the corresponding semitrivial solution is no longer a local minimum of the energy on the Nehari manifold. This change in the local variational geometry produces additional solutions.

When the coupling is sublinear with respect to both variables, the picture becomes particularly interesting: the system possesses a positive least-energy solution for every β > 0, and at least three positive solutions when β is sufficiently small.

Geometry and compactness working together

One aspect of this problem that we find particularly interesting is that the results do not come from a single variational construction.

They arise from the interaction between the geometry of the Nehari manifold, the behavior of the functional near the semitrivial solutions, local and global minimization, mountain-pass arguments, and precise estimates preventing concentration at the critical Sobolev level.

In this sense, moving from subcritical to critical coupling is much more than changing an exponent. It changes the compactness structure of the problem and requires a substantially more delicate variational analysis.

Our paper, Schrödinger systems with critical coupling in Rᴺ, was published in Calculus of Variations and Partial Differential Equations.

Read the paper: https://doi.org/10.1007/s00526-026-03269-6