Classification of singular limits for free boundary and singularly perturbed elliptic problems: the Dancer-Yan spikes revisited

Problem Statement
This paper addresses the classification of singular limits for a free boundary problem arising in plasma physics, specifically modeling plasma equilibria in a Tokamak cross-section. The problem is formulated as a singularly perturbed elliptic equation in dimension $d=2$:
$$\begin{cases} -\Delta v = [v]_+^p & \text{in } \Omega, \\ v = \gamma & \text{on } \partial\Omega, \\ \int_\Omega [v]_+^p = I, \end{cases}$$
where $\Omega \subset \mathbb{R}^2$ is a bounded $C^{2,\beta}$ domain, $p \in (1, +\infty)$, and $I > 0$ represents a fixed total current. The authors investigate the asymptotic behavior of solutions as the total current $I \to +\infty$. In this limit, the "plasma region" $\Omega_+ = \{v > 0\}$ is expected to develop a non-trivial structure, potentially concentrating into "spikes" (regions where $v$ is significantly larger than the boundary value).

While the behavior of such problems in higher dimensions ($d \ge 3$) has been established as a superposition of finitely many "spikes" (specifically Dancer-Yan spikes), the two-dimensional case presents unique difficulties. In $d=2$, the standard limiting global problem $-\Delta w = [w-1]_+^p$ in $\mathbb{R}^2$ with finite mass admits no solutions, preventing the direct application of higher-dimensional blow-up arguments.

Methodology
The authors employ a refined blow-up analysis tailored to the two-dimensional setting, building upon the framework of Dancer and Yan but introducing new rescaling techniques to handle the lack of standard global solutions.

1. Variable Transformation: The problem is reformulated using variables $(\alpha, \psi)$ where $v = \lambda^{1/(p-1)}(\alpha + \lambda \psi)$ and $\lambda = I^{1/q}$ ($q$ being the Hölder conjugate of $p$). This transforms the problem into a singularly perturbed form involving a small parameter $\varepsilon_n \to 0$.
2. Integral Bounds: The analysis relies on two natural integral bounds, denoted (H1) and (H2), which control the $(p-1)$-mass and $p$-mass of the solutions, respectively. These bounds are crucial for ruling out "infinite mass" solutions and ensuring compactness.
3. Refined Rescaling: Unlike the $d \ge 3$ case where local maxima stay bounded away from 1, in $d=2$ the maxima $v_n(x_n)$ converge to 1. The authors introduce a delicate, multi-scale rescaling involving parameters $s_n$ and $\theta_n$ (derived from the Emden equation solution $\phi$) to capture the "microscopic" structure of the spikes.
   • They define a rescaled function $\tilde{v}_n$ and a further normalized function $u_n$ to analyze the profile near local maximizers.
4. Classification of Entire Solutions: A key technical step involves the classification of finite mass solutions to the planar equation $-\Delta w = [w]_+^p$ in $\mathbb{R}^2$. The authors prove that such solutions are either trivial or radial and take a specific form involving the Emden solution $\phi$ inside a disk and a logarithmic decay outside.
5. Extraction of Multiple Spikes: To handle multiple spikes clustering at the same point, the authors use an inductive argument. They prove that two distinct spikes cannot be arbitrarily close relative to their scaling parameters and that the presence of a Type I or Type II spike precludes the existence of a "Fading" spike at the same location.
6. Pohozaev Identity: For the global structure, the authors utilize the Pohozaev identity to derive a constraint on the locations of the spikes, showing they must be critical points of a Kirchhoff-Routh Hamiltonian.

Key Contributions and Results

1. Classification of Singular Limits (Theorem 1.1):
   The authors establish that under the integral bounds (H1) and (H2), the singular limit of solutions falls into one of three categories around any sequence of local maximizers:

   • Vanishing: The solution stays below the threshold 1 in compact subsets.
   • Type I Spikes: The solution converges to a "Dancer-Yan spike" profile. Here, the rescaled function converges to the standard entire solution $w^*$, and the mass is quantized.
   • Type II Spikes: A new phenomenon specific to $d=2$. The rescaled function still converges to $w^*$, but the scaling parameters diverge in a way that the $p$-mass vanishes in the limit, while the $(p-1)$-mass remains quantized.
   • Fading Spikes: The spike amplitude decays so rapidly that it effectively vanishes in the limit, with the solution behaving like a harmonic function perturbed by a small term.

   Crucially, the paper demonstrates that unlike in higher dimensions, not every solution is a simple superposition of Dancer-Yan spikes. The presence of Type II and Fading spikes indicates a richer singular structure in 2D.

2. Global Structure with Dirichlet Conditions (Theorem 1.2):
   By imposing Dirichlet boundary conditions ($v=0$ on $\partial\Omega$) and a non-vanishing condition (NVp) on the $p$-mass, the authors provide a detailed classification for the global solution, while noting that a complete description of the interaction between different spike types clustering at the same point remains an open problem.

   • The singular set $\Sigma$ consists of finitely many interior points.
   • The solution is a superposition of Type I and Type II spikes centered at these points.
   • The $(p-1)$-mass is quantized: it equals $(N_I + N_{II})I_{p-1}$, where $N_I$ and $N_{II}$ are the counts of Type I and Type II spikes.
   • The $p$-mass is not quantized in general; it depends on the specific scaling limits of the Type I spikes.
   • The plasma region $\Omega_{n,+}$ consists of asymptotically round components around the spike centers.
   • The locations of the Type I spikes are critical points of a Kirchhoff-Routh Hamiltonian involving the Green's function of the domain.
   • Note: While the authors successfully rule out "Fading" spikes in the presence of Type I or Type II spikes under these conditions, they explicitly state that a full description of the local interaction when multiple spikes of different types (Type I and Type II) cluster at the same point is currently missing.

3. Application to the Original Plasma Problem (Theorem 1.3):
   The authors translate these results back to the original variables $(v, \gamma)$ for the plasma problem. They show that under specific asymptotic constraints on the parameters $\lambda_n$ and $\alpha_n$, the solutions exhibit the classified spike structures. They provide precise asymptotic formulas for the boundary value $\gamma$ and the total current in terms of the number and type of spikes.

Significance and Claims
The paper claims to provide the first detailed description of the singular limit for this free boundary problem in dimension $d=2$. Its primary significance lies in revealing that the two-dimensional case is fundamentally different from the higher-dimensional case:

• Richness of Structure: The existence of Type II spikes and Fading spikes shows that the solution space is more complex than a simple gluing of standard profiles.
• Quantization Phenomenon: The paper highlights a distinct quantization behavior where the $(p-1)$-mass is quantized (similar to Liouville-type equations), but the $p$-mass is not, challenging the intuition carried over from $d \ge 3$.
• Methodological Advancement: The work introduces a necessary refinement in rescaling techniques to handle the vanishing of the maximum value in 2D, overcoming the obstacle that the standard limiting global problem has no solutions.

The authors explicitly state that a full description of the interaction between multiple spikes of different types clustering at the same point remains an open problem, though they successfully rule out certain configurations (like Fading spikes coexisting with Type I/II spikes) under Dirichlet conditions. They also conjecture that Type II spikes may not exist on convex domains, suggesting a dependence on the domain's geometry.