A Lagrangian approach for prescribed mass solutions of cubic-quintic Schrödinger equations and $L^2$-supercritical problems

We study the existence of radially symmetric solutions of the following nonlinear scalar field equations in ${\mathbb{R}}^N$ ($N\ge 2$): $$(\ast {)}_m-\mathrm{\Delta }u+\mu u=g(u)\text{in }{\mathbb{R}}^N,\frac{1}{2}{\int }_{{\mathbb{R}}^N}{u}^{2}dx=m,$$ 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 $(\ast {)}_m$: $$J_m(\mu ,u)=\frac{1}{2}{\int }_{{\mathbb{R}}^N}\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{2}dx-{\int }_{{\mathbb{R}}^N}G(u)dx+\mu (\frac{1}{2}{\int }_{{\mathbb{R}}^N}{u}^{2}dx-m)\in C^1((0,\mathrm{\infty })\times H_r^1({\mathbb{R}}^N),\mathbb{R})$$ and we give new general existence results through the function: $$b_m:(0,\mathrm{\infty })\to \mathbb{R};\mu \mapsto \text{Mountain Pass minimax value for }(u\mapsto J_m(\mu ,u))\mathrm{.}$$ We will show the existence of solutions of $(\ast {)}_m$ 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].