## Taiwanese Journal of Mathematics

- Taiwanese J. Math.
- Advance publication (2019), 26 pages.

### Infinitely Many Solutions for Sublinear Modified Nonlinear Schrödinger Equations Perturbed from Symmetry

Liang Zhang, Xianhua Tang, and Yi Chen

#### Abstract

In this paper, we consider the existence of infinitely many solutions for the following perturbed modified nonlinear Schrödinger equations \[ \begin{cases} -\Delta u - \Delta(|u|^{\alpha}) |u|^{\alpha-2}u = g(x,u) + h(x,u) &x \in \Omega, \\ u = 0 &x \in \partial \Omega, \end{cases} \] where $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$ ($N \geq 1$) and $\alpha \geq 2$. Under the condition that $g(x,u)$ is sublinear near origin with respect to $u$, we study the effect of non-odd perturbation term $h(x,u)$ which breaks the symmetry of the associated energy functional. With the help of modified Rabinowitz's perturbation method and the truncation method, we prove that this equation possesses a sequence of small negative energy solutions approaching to zero.

#### Article information

**Source**

Taiwanese J. Math., Advance publication (2019), 26 pages.

**Dates**

First available in Project Euclid: 11 October 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.twjm/1539223225

**Digital Object Identifier**

doi:10.11650/tjm/181002

**Subjects**

Primary: 35J20: Variational methods for second-order elliptic equations 35J65: Nonlinear boundary value problems for linear elliptic equations 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]

**Keywords**

broken symmetry infinitely many solutions Rabinowitz's perturbation method modified nonlinear Schrödinger equations

#### Citation

Zhang, Liang; Tang, Xianhua; Chen, Yi. Infinitely Many Solutions for Sublinear Modified Nonlinear Schrödinger Equations Perturbed from Symmetry. Taiwanese J. Math., advance publication, 11 October 2018. doi:10.11650/tjm/181002. https://projecteuclid.org/euclid.twjm/1539223225