Existence and limiting profile of energy ground states for a quasi-linear Schrödinger equations: Mass super-critical case

In any dimension $N\ge 1$, for given mass $a>0$, we look to critical points of the energy functional $$I(u)=\frac{1}{2}{\int }_{{\mathbb{R}}^N}\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{2}dx+{\int }_{{\mathbb{R}}^N}{u}^2\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{2}dx-\frac{1}{p}{\int }_{{\mathbb{R}}^N}\mathrm{\mid }u{\mathrm{\mid }}^{p}dx$$ constrained to the set $${\mathcal{S}}_a=\{u\in X\mathrm{\mid }{\int }_{{\mathbb{R}}^N}\mathrm{\mid }u{\mathrm{\mid }}^{2}dx=a\},$$ where $$X:=\{u\in H^1({\mathbb{R}}^N)\mid {\int }_{{\mathbb{R}}^N}{u}^2\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{2}dx<\mathrm{\infty }\}\mathrm{.}$$ We focus on the mass super-critical case $$4+\frac{4}{N}<p<2\cdot 2^{\ast },where{2}^{\ast }:=\frac{2N}{N-2}forN\ge 3,while{2}^{\ast }:=+\mathrm{\infty }forN=1,2.$$ We explicit a set ${\mathcal{P}}_a\subset {\mathcal{S}}_a$ which contains all the constrained critical points and study the existence of a minimum to the problem \begin{equation*} M_{a}:=\inf_{$\mathcal{P}$_{a}}I(u). \end{equation*} A minimizer of $M_a$ corresponds to an energy ground state. We prove that $M_a$ is achieved for all mass $a>0$ when $1\le N\le 4$. For $N\ge 5$, we find an explicit number $a_0$ such that the existence of minimizer is true if and only if $a\in (0,a_0]$. In the mass super-critical case, the existence of a minimizer to the problem $M_a$, or more generally the existence of a constrained critical point of $I$ on ${\mathcal{S}}_a$, had hitherto only been obtained by assuming that $p\le 2^{\ast }$. In particular, the restriction $N\le 3$ was necessary. We also study the asymptotic behavior of the minimizers to $M_a$ as the mass $a\downarrow 0$, as well as when $a\uparrow a^{\ast }$, where $a^{\ast }=+\mathrm{\infty }$ for $1\le N\le 4$, while $a^{\ast }=a_0$ for $N\ge 5$.