24/10/2024 at 14:30, Room 1B1, Building RM002
In this talk, we use an enhanced Lyapunov-Schmidt reduction method to study a specific class of nonlinear Schrödinger systems with sublinear coupling terms. We establish the existence of infinitely many nonnegative segregated solutions of the following system:
-∆u + K_1(x)u = μ|u|^(p-2)u + β|v|^(p/2)|u|^(p/2-2)u, x ∈ R^N
-∆v + K_2(x)v = ν|v|^(p-2)v + β|u|^(p/2)|v|^(p/2-2)v, x ∈ R^N
where p ∈ (2, 4), K_j(x), j = 1, 2, represent radially symmetric potential functions, μ > 0, ν > 0, and β<0 is the coupling coefficient.
Due to the sublinearity and nonsmooth nature of the coupling terms, traditional reduction methods face challenges. To address these, we propose a novel framework combining variational and perturbation techniques, using a refined inner-outer decomposition approach. Precisely, we construct solutions that concentrate at distinct points near infinity. These solutions also show a 'dead core' phenomenon, unique to this regime, where non-strict positivity is observed.