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Bifurcation and standing wave solutions for a quasilinear Schrödinger equation

Part of: Partial differential equations Spectral theory and eigenvalue problems Elliptic equations and systems

Published online by Cambridge University Press:  27 December 2018

Guowei Dai
Guowei Dai*
Affiliation:
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, PR China ([email protected].)
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Abstract

We use bifurcation and topological methods to investigate the existence/nonexistence and the multiplicity of positive solutions of the following quasilinear Schrödinger equation

$$\left\{ {\matrix{ {-\Delta u-\kappa \Delta \left( {u^2} \right)u = \beta u-\lambda \Phi \left( {u^2} \right)u{\mkern 1mu} {\mkern 1mu} } \hfill & {{\rm in}\;\Omega ,} \hfill \cr {u = 0} \hfill & {{\rm on}\;\partial \Omega } \hfill \cr } } \right.$$
involving sublinear/linear/superlinear nonlinearities at zero or infinity with/without signum condition. In particular, we study the changes in the structure of positive solution with κ as the varying parameter.

Keywords

Bifurcationpositive solutionsQuasilinear Schrödinger equation

MSC classification

Primary: 35B32: Bifurcation
Secondary: 35J60: Nonlinear elliptic equations 35P30: Nonlinear eigenvalue problems, nonlinear spectral theory
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 4 , August 2019 , pp. 939 - 968
Copyright
Copyright © Royal Society of Edinburgh 2018 

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