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

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.

Keywords

• Modified nonlinear Schrodinger equation
• dual approach
• critical point theorem
• multiplicity
• variational methods