Serrin-type Overdetermination in Scaling Limits: Sharp Existence and Asymptotic Behavior of Quasi-linear Schrödinger Energy Ground States

Abstract

This paper establishes optimal existence results and limiting profiles for energy ground states of the quasi-linear Schrödinger equation $$-\hat{I}”u-\hat{I}”(|u{|}^2)u+\hat{I}»u=|u{|}^{p-2}u\text{in }{\mathbb{R}}^N$$ with prescribed mass ${\int }_{{\mathbb{R}}^N}|u{|}^2=a>0$, in the mass-supercritical case $4+\frac{4}{N}<p<2\cdot 2^{\ast }$. Breakthrough in existence theory: For all dimensions $N\ge 1$, we completely resolve the existence problem: For $1\le N\le 4$, ground states exist for all $a>0$. For $N\ge 5$, there exists a sharp threshold $a_0>0$ such that ground states exist if and only if $a\le a_0$. This constitutes the optimal existence theory, crucially removing the restrictive condition $p\le 2^{\ast }$ required in all prior works (which limited results to $N\le 3$). Asymptotic behavior and new phenomena: We provide a complete asymptotic analysis of normalized ground states: As $a\to 0^{+}$, solutions exhibit a novel connection to Serrin-type overdetermined problems. Through a delicate rescaling, profiles converge to the unique positive radial solution of the overdetermined problem (the first such result for quasi-linear equations). As $a\to a^{\ast }$ ($a^{\ast }=\infty$ for $N\le 4$; $a^{\ast }=a_0$ for $N\ge 5$), solutions converge to distinct limiting profiles depending on dimension and nonlinearity. Our methods introduce a new constraint approach and unified variational framework for quasi-linear problems with $L^2$-constraints.

This manuscript is an improved version of the manuscript that was titled 'Existence and limiting profile of energy ground states for a quasi-linear Schrödinger equation: Mass super-critical case.' Note that the title of the document has been changed. This latest version corresponds to the article that will be published in Transactions AMS