Asymptotic behavior of solutions to a planar Hartree equation with isolated singularities

Technical Summary: Asymptotic Behavior of Solutions to a Planar Hartree Equation with Isolated Singularities

Problem Statement
This paper investigates the asymptotic behavior of singular solutions to a planar Hartree-type equation with an isolated singularity at the origin. The primary equation under study is:
$$-\Delta u(x) = \left( \frac{1}{|x|^\alpha} * e^u \right) e^u(x) \quad \text{in } B_1 \setminus \{0\} \subset \mathbb{R}^2,$$
where $\alpha > 0$ and the convolution term is defined as $\frac{1}{|x|^\alpha} * e^u = \int_{B_1 \setminus \{0\}} \frac{e^{u(y)}}{|x-y|^\alpha} dy$. The authors focus on solutions $u \in C^2(B_1 \setminus \{0\})$ satisfying the finite total curvature condition:
$$\int_{B_1 \setminus \{0\}} \left( \frac{1}{|x|^\alpha} * e^u \right) e^u(x) dx < +\infty.$$
The study is motivated by the need to understand singular solutions in geometric problems (such as prescribed Gaussian curvature) and physical models involving nonlocal interactions. While significant results exist for local equations (e.g., $-\Delta u = e^u$) and higher-order Yamabe equations, the nonlocal nature of the Hartree term in two dimensions presents unique challenges, particularly regarding the applicability of ODE-based techniques.

Methodology
The authors employ a Partial Differential Equation (PDE) approach, specifically adapting methods used for higher-order conformal equations with isolated singularities. The core methodology involves:

1. Representation Formula Derivation: Instead of relying on ODE techniques (which fail due to the convolution term) or the maximum principle (which is compromised by nonlocality), the authors establish an integral representation formula for the singular solutions. This is achieved by characterizing the isolated singularities of the Poisson equation and utilizing the finite total curvature condition to control the growth of the solution.
2. Decomposition of Solutions: The solution $u$ is decomposed into three parts: a singular logarithmic term ($b \ln |x|$), a regular harmonic (or polyharmonic) part ($h$), and a potential term ($v$) generated by the nonlocal source.
3. Regularity Analysis: The authors analyze the regularity of the potential term $v$. By establishing that the source term belongs to $L^{p_0}_{loc}$ for some $p_0 > 1$ under specific constraints on the exponent $b$ and $\alpha$, they apply standard elliptic regularity theory (Sobolev embedding and $W^{k,p}$ estimates) to prove that $v$ is locally Hölder continuous.
4. Extension to Higher Dimensions and Orders: The analysis is extended to higher-order Hartree-type equations in dimensions $n \ge 3$:
   • Even Dimensions ($n \ge 4$): For the equation $(-\Delta)^{n/2} u = (\frac{1}{|x|^\alpha} * e^u) e^u$.
   • Odd Dimensions ($n \ge 3$): For the same equation interpreted in the distributional sense, as $(-\Delta)^{n/2}$ becomes a nonlocal integral operator.
   • In higher dimensions, an additional decay condition near the origin, $\int_{B_r \setminus \{0\}} |u(x)| dx = o(r^{n-2})$, is required to eliminate higher-order derivative terms in the representation formula.
5. Generalization: The results are generalized to include a general non-negative coefficient $K(x) \in L^\infty(B_1)$.

Key Contributions and Results

• Asymptotic Representation (Theorem 1.1): For the planar case ($n=2$), the authors prove that any solution $u$ satisfying the finite total curvature condition admits the representation:
  $$u(x) = v(x) + h(x) + b \ln |x|,$$
  where $h$ is a smooth harmonic function, $v$ is a specific integral potential, and $b > -2$ is a constant. Furthermore, if $\alpha \in (0, 2)$ and $b > \frac{\alpha-4}{2}$, the solution behaves as $u(x) = \phi(x) + b \ln |x|$ near the origin, where $\phi$ is a Hölder continuous function.

• Higher-Order Even Dimensions (Theorem 1.2): For even $n \ge 4$, under the finite total curvature and the additional decay condition, a similar representation holds:
  $$u(x) = v(x) + h(x) + b \ln |x|,$$
  with $h$ being polyharmonic ($(-\Delta)^{n/2} h = 0$) and $b > -n$. The Hölder continuity of the remainder term is established under the condition $b > \frac{\alpha-2n}{2}$.

• Higher-Order Odd Dimensions (Theorem 1.4): For odd $n \ge 3$, treating the equation in the distributional sense, the same asymptotic structure is derived. The authors define solutions in the space $L^{n/2}(\mathbb{R}^n) \cap C^n(B_1 \setminus \{0\})$ and prove the existence of the logarithmic singularity and Hölder continuous remainder under the same constraints on $b$.

• Validity Range of the Singularity Coefficient: The paper rigorously establishes that the condition $b > \frac{\alpha-2n}{2}$ is necessary for the finite total curvature condition to hold. If $b \le \frac{\alpha-2n}{2}$, the integral of the nonlocal term diverges, contradicting the initial assumption.

• General Coefficient Extension (Remark 1.7): The authors demonstrate that these results extend to equations with a general non-negative bounded coefficient $K(x)$, modifying the integral representation of $v$ accordingly.

Significance and Claims
The paper claims to fill a gap in the literature regarding Hartree-type equations with isolated singularities in two dimensions. While classification and asymptotic results exist for local equations (like the prescribed scalar curvature equation) and higher-order local Yamabe equations, the nonlocal nature of the Hartree term in $\mathbb{R}^2$ had not been sufficiently addressed.

The authors highlight that their PDE-based approach successfully bypasses the limitations of ODE methods (which cannot handle the convolution term) and the breakdown of the maximum principle due to nonlocality. By establishing a representation formula and proving the Hölder continuity of the nonlocal potential, they provide a complete description of the asymptotic behavior of singular solutions near the origin. The work also unifies the treatment of even and odd dimensions for higher-order Hartree equations, extending known results from local conformal geometry to nonlocal settings.