Sign-changing solutions for a parameter-dependent quasilinear equation

Jiaquan Liu; Xiangqing Liu; Zhi-Qiang Wang

doi:10.3934/dcdss.2020454

Article Contents

2021, Volume 14, Issue 5: 1779-1799. Doi: 10.3934/dcdss.2020454

This issue Previous Article Compactness results for linearly perturbed Yamabe problem on manifolds with boundary Next Article

Sign-changing solutions for a parameter-dependent quasilinear equation

1.
LMAM, School of Mathematical Science, Peking University, Beijing 100871, China
2.
Department of Mathematics, Yunnan Normal University, Kunming 650500, China
3.
Department of Mathematics and Statistics, Utah State University, Logan, UT 84322, USA

^* Corresponding author: Xiangqing Liu, Zhi-Qiang Wang
^* Corresponding author: Xiangqing Liu, Zhi-Qiang Wang

Received: January 31, 2020

Revised: June 30, 2020

Early access: November 2020

Published: May 2021

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Abstract

We consider quasilinear elliptic equations, including the following Modified Nonlinear Schrödinger Equation as a special example:

$ \begin{equation*} \left\{ \begin{aligned} &\Delta u+\frac{1}{2}u\Delta u^2+\lambda |u|^{r-2}u = 0, \ \ \ \text{in}\,\,\Omega,\\ &u = 0\quad\text{on}\,\,\partial\Omega, \end{aligned} \right. \end{equation*} $

where $ \Omega\subset\mathbb{R}^N(N\geq3) $ is a bounded domain with smooth boundary, $ \lambda>0,\, r\in(2,4) $. We prove as $ \lambda $ becomes large the existence of more and more sign-changing solutions of both positive and negative energies.

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Mathematics Subject Classification: 35B20, 35H30.

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