## Topological Methods in Nonlinear Analysis

### Infinitely many solutions for quasilinear Schrödinger equations under broken symmetry situation

#### Abstract

In this paper, we study the existence of infinitely many solutions for the quasilinear Schrödinger equations $$\begin{cases} -\Delta u-\Delta(|u|^{\alpha})|u|^{\alpha-2}u=g(x,u)+h(x,u) &\text{for } x\in \Omega,\\ u=0 &\text{for } x\in \partial\Omega, \end{cases}$$ where $\alpha\geq 2$, $g, h\in C(\Omega\times \mathbb{R}, \mathbb{R})$. When $g$ is of superlinear growth at infinity in $u$ and $h$ is not odd in $u$, the existence of infinitely many solutions is proved in spite of the lack of the symmetry of this problem, by using the dual approach and Rabinowitz perturbation method. Our results generalize some known results and are new even in the symmetric situation.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 48, Number 2 (2016), 539-554.

Dates
First available in Project Euclid: 21 December 2016

https://projecteuclid.org/euclid.tmna/1482289228

Digital Object Identifier
doi:10.12775/TMNA.2016.057

Mathematical Reviews number (MathSciNet)
MR3642772

Zentralblatt MATH identifier
1362.35020

#### Citation

Zhang, Liang; Tang, Xianhua; Chen, Yi. Infinitely many solutions for quasilinear Schrödinger equations under broken symmetry situation. Topol. Methods Nonlinear Anal. 48 (2016), no. 2, 539--554. doi:10.12775/TMNA.2016.057. https://projecteuclid.org/euclid.tmna/1482289228

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