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

TL;DR

The paper addresses the existence and asymptotic profiles of energy ground states for a quasi-linear Schrödinger equation under a prescribed mass in the mass-supercritical range $p\in\bigl(4+\frac{4}{N},2\cdot 2^*\bigr)$. It develops a variational framework based on a Pohozaev-type constraint and a relaxed problem to overcome non-differentiability and dimension-specific difficulties, proving existence for all $a>0$ when $1\le N\le 4$ and a threshold phenomenon for $N\ge5$ with a critical mass $a_0$. The paper further analyzes the limiting behavior as $a\to0^+$ and as $a\to a^*$ (with $a^*=\infty$ for $N\le4$ and $a^*=a_0$ for $N\ge5$), identifying convergent profiles: in many regimes the limits solve reduced equations or overdetermined problems on balls, and the Lagrange multipliers tend to $0$ or $+\infty$ accordingly. These results extend prior work by removing restrictions on the nonlinearity exponent and by providing a detailed description of the limiting energies and ground-state shapes across dimensions.

Abstract

In any dimension $N \geq 1$, for given mass $a>0$, we look to critical points of the energy functional $$ I(u) = \frac{1}{2}\int_{\mathbb{R}^N}|\nabla u|^2 dx + \int_{\mathbb{R}^N}u^2|\nabla u|^2 dx - \frac{1}{p}\int_{\mathbb{R}^N}|u|^p dx$$ constrained to the set $$\mathcal{S}_a=\{ u \in X | \int_{\mathbb{R}^N}| u|^2 dx = a\},$$ where $$ X:=\left\{u \in H^1(\mathbb{R}^N)\Big| \int_{\mathbb{R}^N} u^2|\nabla u|^2 dx <\infty\right\}. $$ We focus on the mass super-critical case $$4+\frac{4}{N}<p<2\cdot 2^*, \quad \mbox{where } 2^*:=\frac{2N}{N-2} \quad \mbox{for } N\geq 3, \quad \mbox{while } 2^*:=+\infty \quad \mbox{for } N=1,2.$$ 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 $1\leq N\leq 4$. For $N\geq 5$, 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 $p \leq 2^*$. In particular, the restriction $N \leq 3$ was necessary. We also study the asymptotic behavior of the minimizers to $M_a$ as the mass $a \downarrow 0$, as well as when $a \uparrow a^*$, where $a^*=+\infty$ for $1\leq N\leq 4$, while $a^*=a_0$ for $N\geq 5$.

Theorems & Definitions (79)

• Theorem 2.1
• Proposition 2.2
• Theorem 2.3
• Remark 2.4
• Remark 2.5
• Theorem 2.6
• Theorem 2.7
• Theorem 2.8
• Theorem 3.1
• Lemma 4.1