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Least Energy Nodal Solutions for a Defocusing Schrödinger Equation with Supercritical Exponent

Part of: Elliptic equations and systems Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  16 August 2018

Minbo Yang ,
Carlos Alberto Santos  and
Jiazheng Zhou
Minbo Yang
Affiliation:
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, People's Republic of China (mbyang@zjnu.edu.cn)
Carlos Alberto Santos*
Affiliation:
Universidade de Brasília, Departamento de Matemática, 70910-900, Brasília DF, Brazil (csantos@unb.br; jiazzheng@gmail.com)
Jiazheng Zhou
Affiliation:
Universidade de Brasília, Departamento de Matemática, 70910-900, Brasília DF, Brazil (csantos@unb.br; jiazzheng@gmail.com)
*
*Corresponding author.
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Abstract

In this paper we consider the existence of least energy nodal solution for the defocusing quasilinear Schrödinger equation

$$-\Delta u - u \Delta u^2 + V(x)u = a(x)[g(u) + \lambda \vert u \vert ^{p-2}u] \hbox{in} {\open R}^N,$$
where λ≥0 is a real parameter, V(x) is a non-vanishing function, a(x) can be a vanishing positive function at infinity, the nonlinearity g(u) is of subcritical growth, the exponent p≥22*, and N≥3. The proof is based on a dual argument on Nehari manifold by employing a deformation argument and an $L</italic>^{\infty}({\open R}^{N})$-estimative.

Keywords

quasilinear Schrödinger equationleast energy nodal solutionssupercritical growthMiranda's Theorem

MSC classification

Primary: 35J20: Variational methods for second-order elliptic equations
Secondary: 35J60: Nonlinear elliptic equations 35Q55: NLS-like equations (nonlinear Schrödinger)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 1 , February 2019 , pp. 1 - 23
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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