Electron. J. Differential Equations, Vol. 2018 (2018), No. 147, pp. 1-15.

Existence of infinitely solutions for a modified nonlinear Schrodinger equation via dual approach

Xinguang Zhang, Lishan Liu, Yonghong Wu, Yujun Cui

Abstract:
In this article, we focus on the existence of infinitely many weak solutions for the modified nonlinear Schrodinger equation
$$ -\Delta u+V(x) u-[\Delta(1+u^2)^{\alpha/2}]\frac{\alpha u}{2(1+u^2) ^{\frac{2-\alpha}2}}=f(x,u),\quad \text{in } \mathbb{R}^N, $$
where $1\leq\alpha<2$, $f \in C(\mathbb{R}^N \times \mathbb{R}, \mathbb{R})$. By using a symmetric mountain pass theorem and dual approach, we prove that the above equation has infinitely many high energy solutions.

Submitted March 8, 2018. Published July 31, 2018.
Math Subject Classifications: 35J50, 35J92.
Key Words: Modified nonlinear Schrodinger equation; dual approach; critical point theorem; multiplicity; variational methods.

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Xinguang Zhang
School of Mathematical and Informational Sciences
Yantai University
Yantai 264005, Shandong, China
email: xinguang.zhang@curtin.edu.au
Lishan Liu
School of Mathematical Sciences
Qufu Normal University
Qufu 273165, Shandong, China
email: mathlls@163.com
Yonghong Wu
Department of Mathematics and Statistics
Curtin University of Technology
Perth, WA 6845, Australia
email: y.wu@curtin.edu.au
Yujun Cui
Department of Mathematics
Shandong University of Science and Technology
Qingdao, 266590, Shandong, China
email: cyj720201@163.com

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