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

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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