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

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

1. Problem Formulation
This paper investigates the existence and asymptotic behavior of energy ground states for the quasi-linear Schrödinger equation in $\mathbb{R}^N$:
$$-\Delta u - \Delta(|u|^2)u + \lambda u = |u|^{p-2}u$$
subject to a prescribed mass (normalization) constraint:
$$\int_{\mathbb{R}^N} |u|^2 dx = a > 0.$$
The study focuses on the mass-supercritical regime, defined by the exponent range $4 + \frac{4}{N} < p < 2 \cdot 2^*$, where $2^* = \frac{2N}{N-2}$ is the critical Sobolev exponent (with $2^* = \infty$ for $N=1,2$). In this regime, the associated energy functional is unbounded from below on the constraint set, making the existence of minimizers non-trivial. The frequency $\lambda$ appears as a Lagrange multiplier determined by the constraint.

2. Methodology
The authors address the non-differentiability of the energy functional $I(u)$ in the natural space $X$ (due to the term $u\Delta(|u|^2)$) and the lack of compactness in the supercritical case through the following strategies:

• Relaxed Minimization Approach: Instead of minimizing directly on the intersection of the constraint sphere $S_a$ and the Pohozaev manifold $\mathcal{P}$, the authors introduce a relaxed problem on the set $D_a = \{u \in X : \|u\|_2^2 \leq a\}$. They define $M_a = \inf_{u \in \mathcal{P} \cap D_a} I(u)$.
• Variational Framework: They prove that minimizers of $M_a$ are critical points of the energy restricted to $D_a$. A key step involves showing that for specific dimensions and mass ranges, these minimizers lie strictly on the boundary $\partial D_a = S_a$, thereby solving the original normalized problem.
• Dual Transformation: To handle the quasi-linear structure, the authors utilize a change of variables $u = \phi(v)$, transforming the problem into a semi-linear equation for $v$. This allows the application of standard elliptic regularity and uniqueness results for radial solutions.
• Asymptotic Analysis via Rescaling: For the limiting behaviors ($a \to 0^+$ and $a \to a^*$), the authors employ delicate rescaling techniques. As $a \to 0^+$, the solutions are rescaled to converge to a unique positive radial solution of a specific Serrin-type overdetermined problem (a free boundary problem with Dirichlet-Neumann conditions). As $a \to a^*$, the analysis relies on the convergence to solutions of the zero-mass equation or standard semi-linear limits.

3. Key Contributions and Results

• Optimal Existence Theory (Removing the $p \leq 2^*$ Barrier):
  Prior works (e.g., [30, 43, 55]) required the restrictive assumption $p \leq 2^*$, which limited existence results to dimensions $N \leq 3$. This paper establishes existence for the complete range $4 + \frac{4}{N} < p < 2 \cdot 2^*$ for all dimensions $N \geq 1$.

  • Case $1 \leq N \leq 4$: Energy ground states exist for all prescribed masses $a > 0$. The associated Lagrange multiplier $\lambda$ is strictly positive.
  • Case $N \geq 5$: A sharp threshold $a_0 > 0$ exists. Ground states exist if and only if $a \leq a_0$.
    • For $a < a_0$, a unique (up to translation/sign) positive radial ground state exists with $\lambda > 0$.
    • For $a = a_0$, the ground state is the unique positive radial solution to the zero-mass equation (where $\lambda = 0$).
    • For $a > a_0$, no energy ground state exists.

• Asymptotic Behavior as $a \to 0^+$:
  The paper provides a novel connection between the quasi-linear ground states and Serrin-type overdetermined problems. As the mass $a$ approaches zero, the rescaled profiles of the ground states converge to the unique positive radial solution of:
  $$\begin{cases} -\Delta \tilde{v} = -\frac{\sqrt{2}}{2} + 2^{\frac{p-4}{4}} \tilde{v}^{\frac{p-2}{2}} & \text{in } \Omega, \\ \tilde{v} > 0 & \text{in } \Omega, \\ \tilde{v} = \frac{\partial \tilde{v}}{\partial \nu} = 0 & \text{on } \partial \Omega, \end{cases}$$
  where $\Omega$ is a ball. This constitutes the first such result for quasi-linear equations. The Lagrange multiplier $\lambda$ diverges to $+\infty$ in this limit.

• Asymptotic Behavior as $a \to a^*$:

  • For $N \leq 4$ (where $a^* = \infty$) and $N \geq 5$ (where $a^* = a_0$), $\lambda \to 0^+$.
  • Depending on the dimension and the exponent $p$, the rescaled solutions converge to:
    • The unique positive radial solution of the standard semi-linear equation $-\Delta W + W = W^{p-1}$ (for $N=2,3$ and sub-critical $p$).
    • The Talenti bubble (for $N=3, p=2^*$).
    • The unique positive radial solution of the zero-mass quasi-linear equation (for $N \geq 3$ with super-critical $p$).

4. Significance and Claims
The paper claims to provide the optimal existence theory for normalized ground states of this class of quasi-linear equations. The primary breakthrough is the removal of the long-standing restriction $p \leq 2^*$, which previously confined existence results to low dimensions ($N \leq 3$). By establishing existence for the full supercritical range $p < 2 \cdot 2^*$ across all dimensions, the authors resolve the existence problem completely.

Furthermore, the work introduces a unified variational framework using a relaxed constraint approach that avoids the differentiability issues inherent in the quasi-linear functional. The identification of the Serrin-type overdetermined problem as the limiting profile for vanishing mass represents a new phenomenon in the asymptotic analysis of quasi-linear Schrödinger equations, linking the behavior of ground states to free boundary problems.

The authors note that while the variational characterization suggests instability (blow-up) for the associated standing waves, a definitive conclusion on orbital stability remains open due to the lack of a local well-posedness theory for the equation in the energy space $X$. However, the paper establishes that the energy ground states coincide with action ground states (minimizers of the action functional for fixed $\lambda$) due to the uniqueness of positive solutions for fixed $\lambda > 0$.