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

In this study, we investigate the existence of solutions \((\lambda , u) \in \mathbb {R} \times H^1(\mathbb {R}^N)\) $(\lambda ,u)\in \mathbb{R}\times H^1({\mathbb{R}}^N)$ to the Schrödinger equation \(\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+V(x)u+\lambda u=|u|^{p-2}u,\quad x\in \mathbb {R}^{N},\\ \int _{\mathbb {R}^N}|u|^2=a, \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}-\mathrm{\Delta }u+V(x)u+\lambda u={|u|}^{p-2}u,x\in {\mathbb{R}}^N,\hfill \\ {\int }_{{\mathbb{R}}^N}{|u|}^2=a,\hfill \end{array}\right)\end{array}$$ where \(N\ge 2\) $N\ge 2$ , \(a>0\) $a>0$ is a constant and p satisfies \(2+4/N<p<+\infty \) $2+4/N<p<+\infty$ . The potential V satisfies the condition that the operator \(-\Delta +V\) $-\mathrm{\Delta }+V$ contains infinitely many isolated eigenvalues with an accumulation point. We prove that this equation has a sequence of solutions \(\{(\lambda _m, u_m)\}\) $\{({\lambda }_m,u_m)\}$ such that \(\Vert u_m\Vert _{L^\infty (\mathbb {R}^N)}\rightarrow 0\) $\Vert u_m{\Vert }_{L^{\infty }({\mathbb{R}}^N)}\to 0$ as \(m\rightarrow \infty \) $m\to \infty$ . The proof is provided by establishing a new critical point theorem without the typical Palais–Smale condition.