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Journal of Fixed Point Theory and Applications

, Volume 19, Issue 1, pp 987–1010 | Cite as

Multiplicity of semiclassical solutions to nonlinear Schrödinger equations

  • Yanheng Ding
  • Juncheng Wei
Article

Abstract

We study the following nonlinear Schrödinger equations:
$$\begin{aligned} -\varepsilon ^2\Delta w + V(x)w = W(x)|w|^{p-2}w \end{aligned}$$
(0.1)
and
$$\begin{aligned} -\varepsilon ^2\Delta w + V(x)w = W_1(x)|w|^{p-2}w + W_2(x)|w|^{2^*-2}w \end{aligned}$$
(0.2)
for \(x\in \mathbb {R}^N\), where \(p\in (2, 2^*)\), \(2^*=2N/(N-2)\) if \(N>2\) and \(=\infty \) if \(N=2\) (and (0.2) is considered just for \(N\ge 3\)), \(\min V>0\) and \(\inf W>0\). Under certain assumptions, we study the existence and concentration phenomena of semiclassical positive ground states, and multiplicity of solutions including at least 1 pair of sign-changing ones for (0.1) and (0.2) with simultaneously linear and nonlinear potentials.

Keywords

Schrödinger equation critical nonlinearities semiclassical solution multiplicity concentration 

Mathematics Subject Classification

58E05 58E50 

Notes

Acknowledgements

The authors thank the reviewer for the suggestions which improve the paper much. The work was supported by the CAS-Croucher Joint Laboratory. Ding was also partly supported by the National Science Foundation of China (NSFC11331010, 11571146).

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Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.Institute of Mathematics, Academy of Mathematics and Systems ScienceUniversity of Chinese Academy of Sciences, CASBeijingPeople’s Republic of China
  2. 2.Department of MathematicsThe Chinese University of Hong KongShatinHong Kong
  3. 3.Department of MathematicsUniversity of British ColumbiaVancouverCanada

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