Serrin-type Overdetermination in Scaling Limits: Sharp Existence and Asymptotic Behavior of Quasi-linear Schr\"odinger Energy Ground States

This paper establishes optimal existence results and limiting profiles for energy ground states of the quasi-linear Schr\"odinger equation $$ -\Delta u - \Delta(|u|^{2})u + \lambda u = |u|^{p-2}u \quad \text{in } \mathbb{R}^N $$ with prescribed mass $\int_{\mathbb{R}^N}|u|^2 = a > 0$, in the mass-supercritical case $4 + \frac{4}{N} < p < 2 \cdot 2^*$. Breakthrough in existence theory: For all dimensions $N \geq 1$, we completely resolve the existence problem: For $1 \leq N \leq 4$, ground states exist for all $a > 0$. For $N \geq 5$, there exists a sharp threshold $a_0 > 0$ such that ground states exist if and only if $a \leq a_0$. This constitutes the optimal existence theory, crucially removing the restrictive condition $p \leq 2^*$ required in all prior works (which limited results to $N \leq 3$). 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 $a \to a^*$ ($a^* = \infty$ for $N \leq 4$; $a^* = a_0$ for $N \geq 5$), 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.