We study normalized solutions (μ, u) ∈ R × H1(RN) to nonlinear Schrödinger equations −Δu + μu = g(u) in R^N, \int^R^N u^2dx = m, where N ≥ 2 and the mass m > 0 is given. Here, g has an L2-critical growth, both at the origin and at infinity, that is, g(s) ∼ |s|p−1s as s ∼ 0 and s ∼ ∞, where p = 1 + 4 N . We continue the analysis started in [11], where we found two (possibly distinct) minimax values b ≤ 0 ≤ b of the Lagrangian functional. In this paper, we furnish explicit examples of g satisfying b < 0 < b, b = 0 < b, and b < 0 = b; notice that b = 0 = b in the power case g(t) = |t|p−1t. Moreover, we deal with the existence and non-existence of a solution with minimal energy. Finally, we discuss the assumptions required on g to obtain the existence of a positive solution for perturbations of g.
NORMALIZED GROUND STATES FOR NLS EQUATIONS WITH MASS CRITICAL NONLINEARITIES
Cingolani Silvia
;Gallo Marco;
2025-01-01
Abstract
We study normalized solutions (μ, u) ∈ R × H1(RN) to nonlinear Schrödinger equations −Δu + μu = g(u) in R^N, \int^R^N u^2dx = m, where N ≥ 2 and the mass m > 0 is given. Here, g has an L2-critical growth, both at the origin and at infinity, that is, g(s) ∼ |s|p−1s as s ∼ 0 and s ∼ ∞, where p = 1 + 4 N . We continue the analysis started in [11], where we found two (possibly distinct) minimax values b ≤ 0 ≤ b of the Lagrangian functional. In this paper, we furnish explicit examples of g satisfying b < 0 < b, b = 0 < b, and b < 0 = b; notice that b = 0 = b in the power case g(t) = |t|p−1t. Moreover, we deal with the existence and non-existence of a solution with minimal energy. Finally, we discuss the assumptions required on g to obtain the existence of a positive solution for perturbations of g.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
