New critical point theorem and infinitely many normalized small-magnitude solutions of mass supercritical Schrödinger equations

Abstract

In this study, we investigate the existence of solutions \((\lambda , u) \in \mathbb {R} \times H^1(\mathbb {R}^N)\) to the Schrödinger equation

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+V(x)u+\lambda u=|u|^{p-2}u,\quad x\in \mathbb {R}^{N},\\ \int _{\mathbb {R}^N}|u|^2=a, \end{array} \right. \end{aligned}$$

where \(N\ge 2\), \(a>0\) is a constant and p satisfies \(2+4/N<p<+\infty \). The potential V satisfies the condition that the operator \(-\Delta +V\) contains infinitely many isolated eigenvalues with an accumulation point. We prove that this equation has a sequence of solutions \(\{(\lambda _m, u_m)\}\) such that \(\Vert u_m\Vert _{L^\infty (\mathbb {R}^N)}\rightarrow 0\) as \(m\rightarrow \infty \). The proof is provided by establishing a new critical point theorem without the typical Palais–Smale condition.