Infinitely many solutions for Schrödinger equations with Hardy potential and Berestycki-Lions conditions

Abstract

In this article, we investigate the following Schrödinger equation: − Δ u − μ ∣ x ∣ 2 u = g ( u ) in R N , -\Delta u-\frac{\mu }{{| x| }^{2}}u=g\left(u)\hspace{1em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{N}, where N ≥ 3 N\ge 3 , μ ∣ x ∣ 2 \frac{\mu }{{| x| }^{2}} is called the Hardy potential and g g satisfies Berestycki-Lions conditions. If 0 < μ < ( N − 2 ) 2 4 0\lt \mu \lt \frac{{\left(N-2)}^{2}}{4} , we will take symmetric mountain pass approaches to prove the existence of infinitely many solutions of this problem.