Existence of Semiclassical Bound States of Nonlinear Schrödinger Equations with Potentials of the Class (V)_a

Stability and instability of standing waves for nonlinear Schrödinger equations

“…Also, there have been several papers on the stability properties of standing waves for a nonlinear Schrödinger equation with a potential which has a small parameter h > 0 (see, Oh [39], Example C of Section 6 in Grillakis, Shatah and Strauss [28]). …”

Entire solutions of multivalued nonlinear Schrödinger equations in Sobolev spaces with variable exponent

“…More precisely, it is assumed that V belongs to the class (V a ) (for some a) introduced in Kato [22]. Taking γ > 0 and > 0 sufficiently small and using a Lyapunov-Schmidt type reduction, Oh [31] proved the existence of a standing wave solution of Problem (1), that is, a solution of the form…”

“…In the study of Eq. (1), Oh [31] supposed that the potential V is bounded and possesses a non-degenerate critical point at x = 0. More precisely, it is assumed that V belongs to the class (V a ) (for some a) introduced in Kato [22].…”

Multiscale-bump standing waves with a critical frequency for nonlinear Schrödinger equations

“…Let us point out that the result in Theorem 1.2,which generalizes the corresponding results of Oh in [28] and [29] to nonnegative potentials (a uniform positive lower bound is needed in [28] and [29]), is new. The proof of this result does not follow the reduction procedure in a standard way, because u ε cannot control the L p norm of u uniformly if Z is not empty.…”

“…The solutions concentrate near some critical points of W (x) as → 0. Their method, based on an interesting Lyapunov-Schmidt finite dimensional reduction, was extended by Oh in [28,29] to include a similar result in higher dimensions, provided 1 < p < N +2 N −2 . Other existence results for positive solutions of problem (1.2) under the condition inf x∈R N W (x) > E can be found in [1,2,3,10,12,13,14,15,16,17,20,22,24,25,27,31,32].…”