A Lagrangian approach for prescribed mass solutions of cubic-quintic Schr\"odinger equations and $L^2$-supercritical problems

Abstract

We study the existence of radially symmetric solutions of the following nonlinear scalar field equations in $\mathbb R^N$ ($N \geq 2$): $$ (*)_m \quad - \Delta u + \mu u = g(u) \quad \text{in}\ {\mathbb R}^N, \quad {1\over 2} \int_{{\mathbb R}^N} u^2\, dx = m,$$ where $g(s) \in C({\mathbb R},{\mathbb R})$, $m > 0$ and $\mu \in {\mathbb R}$ is an unknown Lagrangian multiplier. We take an approach using a Lagrangian formulation of $(*)_m$: $$J_m(\mu,u)={1\over 2}\int_{{\mathbb R}^N} |\nabla u|^2\,dx -\int_{{\mathbb R}^N} G(u)\,dx +\mu\left({1\over 2}\int_{{\mathbb R}^N} u^2\, dx-m\right) \in C^1((0,\infty)\times H_r^1({\mathbb R}^N), {\mathbb R})$$ and we give new general existence results through the function: $$ b_m:\, (0,\infty) \to {\mathbb R};\ \mu \mapsto \text{Mountain Pass minimax value for}\ (u\mapsto J_m(\mu,u)).$$ We will show the existence of solutions of $(*)_m$ related to local minima and local maxima of $b_m(\mu)$. As applications, we study cubic-quintic type equations and $L^2$-supercritical problems. In particular, when $N=2,3$, we show new existence results of normalized solutions without assuming global Ambrosetti-Rabinowitz type conditions, which partially improve the preceding results due to Jeanjean [24] and Jeanjean-Lu [26, 28].