Bound states of nonlinear Schrödinger equations with magnetic fields

Abstract

We study bound states of the following nonlinear Schrödinger equation in the presence of a magnetic field:We prove that if V is bounded below with the set {x ∈ R N : V (x) < b} = ∅ having finite measure for some b > 0, inf V ≤ 0, and g satisfies some growth conditions, then for any integer m whenh > 0 is sufficiently small the problem has m geometrically different solutions.

Fractional NLS equations with magnetic field, critical frequency and critical growth

“…(1.3) where ε = , V (x) = 2m(W (x) − ω), f = 2mg and (1.4) ε i ∇ − A(x) 2 u = −ε 2 ∆u − 2ε i A(x) · ∇u + |A(x)| 2 u − ε i divA(x)u. See [14] and the references cited therein for recent results in this direction (see also [31]). Similarly, we could derive the fractional version of (1.3) as A = 0 and ε = 1, which is a fundamental equation of fractional Quantum Mechanics in the study of particles on stochastic fields modeled by Lévy processes, see [23].…”

“…Motivated by the above works, especially by [14,15], we are interested in critical fractional Schrödinger equations with the magnetic field and the critical frequency case in the sense that min R N V = 0. It is worth mentioning that the study of fractional Schrödinger equations with the critical frequency was first investigated by Byeon and Wang [4,5].…”

Variational methods for strongly indefinite problems

“…(g 3 ) ∃ a 2 > 0, 2 < p < 2 * 使得 |g(x, u)| a 2 (|u| p−1 + |u| q−1 ), ∀ (x, u). 定理 31 [15] 这方面我们另外的工作参见文献 [46][47][48][49][50][51][52][53] 等.…”

A monotone property of the ground state energy to the scalar field equation and applications

“…A well‐known result, proved in by the variational method, is that if the potential has a global minimum point , then the spike of the positive ground state of , that is, the maximum point of the ground state, will converge to as . Some further studies about the existence, multiplicity and qualitative properties of semi‐classical states as spiked solutions (positive or sign changing, single peak or multi‐peaks) with various types of concentration behaviors have been established, by the global variational method or the Lyapunov–Schmidt reduction method, under various assumptions on the potential , which can be seen in and the references therein. In particular, it has been proved by the variational methods in that if the potential has a global minimum point , then the two spikes of the least energy sign‐changing solution of , that is, the maximum point and the minimum point of the least energy sign‐changing solution, will both converge to as .…”