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Manuscripta Mathematica

, Volume 140, Issue 1–2, pp 51–82 | Cite as

Semiclassical solutions of Schrödinger equations with magnetic fields and critical nonlinearities

  • Yanheng Ding
  • Xiaoying Liu
Article

Abstract

We study the following nonlinear Schrödinger equations
$$\begin{array}{lll}(-i\varepsilon\nabla+A(x))^2 w + V(x)w = W(x)g(|w|)w; \quad \quad \quad \quad \quad \quad \quad \quad \quad (0.1)\\(-i\varepsilon\nabla+A(x))^2 w + V(x)w = W(x)\left(g(|w|)+|w|^{2^*-2}\right)w,\quad \quad \quad\,\,(0.2)\end{array}$$
for \({w \in H^1\left( \mathbb{R}^N, \mathbb{C} \right)}\) , where g(|w|)w is super linear and subcritical, 2* = 2N/(N − 2) if N > 2 and =  if N = 2, min V > 0 and inf W > 0. Under proper assumptions we explore the existence and concentration phenomena of semiclassical solutions of (0.1). The most interesting result obtained here refers to the critical case. We establish the existence and describe the concentration of semiclassical ground states of (0.2) provided either min Vτ 0 for some τ0 > 0, or \({\max W > \kappa_{0}}\) for some \({\kappa_0 > 0}\) .

Mathematics Subject Classification (2000)

58E05 58E50 

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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Institute of Mathematics, AMSSChinese Academy of SciencesBeijingPeople’s Republic of China
  2. 2.Department of MathematicsXuzhou Normal UniversityXuzhouPeople’s Republic of China

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