Singular semilinear elliptic equations in nondivergence form

Technical Summary: Singular Semilinear Elliptic Equations in Nondivergence Form

Problem Statement
This paper investigates the existence and uniqueness of solutions to the singular semilinear Dirichlet problem:
$$\begin{cases} -Pu = \frac{f}{u^\gamma} & \text{in } \Omega, \\ u \equiv 0 & \text{on } \partial\Omega, \end{cases}$$
where $\Omega \subset \mathbb{R}^d$ ($d \geq 2$) is a bounded domain, $f \in L^1(\Omega)$ is nonnegative and nontrivial, and $\gamma > 0$. The operator $P$ is a second-order uniformly elliptic differential operator in nondivergence form:
$$Pu(x) = \sum_{i,j=1}^d a_{ij}(x) \frac{\partial^2 u}{\partial x_i \partial x_j},$$
with a symmetric matrix $A(x) = [a_{ij}(x)]$ satisfying uniform ellipticity conditions. The singularity arises from the term $u^{-\gamma}$, which is undefined at $u=0$. The boundary condition is interpreted in the sense of a vanishing trace near the boundary: $\lim_{\epsilon \to 0} \frac{1}{\epsilon} \int_{\Omega_\epsilon} |u| = 0$, where $\Omega_\epsilon = \{x \in \Omega : \delta_\Omega(x) < \epsilon\}$.

Methodology and Technical Framework
The authors employ a combination of approximation techniques, nonlinear inequalities, and Green function estimates. The proofs distinguish between two notions of solutions based on the regularity of the coefficients $A$ and the solution $u$:

1. Weak Solutions: Defined for $A \in C^1(\Omega)$, where the equation holds in the sense of distributions using the formulation $\langle -Pu, \phi \rangle = \int_\Omega (A\nabla u \cdot \nabla \phi + (\text{div} A \cdot \nabla u)\phi)$.
2. Distributional Solutions: Defined for $A \in C^2(\Omega)$, where the equation holds via the adjoint operator $P^*$: $\langle -Pu, \phi \rangle = -\int_\Omega u P^* \phi$.

Key Contributions and Results

1. Existence and Regularity (Theorem 1.1)
Under the assumptions that $\Omega \in C^{1,1}$, $A \in C^1(\Omega)$, and a specific smallness condition on the drift term $\text{div} A$ (inequality 1.5), the authors prove the existence of a weak solution $u \in W^{1,1}_{\text{loc}}(\Omega) \cap L^1(\Omega)$.

• Regularity: The solution exhibits specific Sobolev regularity depending on $\gamma$:
  • $u \in H^1_{\text{loc}}(\Omega)$ for $\gamma \geq 2$.
  • $u \in H^1_{\text{loc}}(\Omega) \cap W^{1,q}_0(\Omega)$ for $\gamma \in (1, 2)$ and $1 \leq q < 2/\gamma$.
  • $u \in H^1_0(\Omega)$ for $\gamma = 1$.
  • $u \in W^{1,q}_0(\Omega)$ for $\gamma \in (0, 1)$ with a specific exponent $q$.
  • Crucially, $u^{(\gamma+1)/2} \in H^1_0(\Omega)$ for all $\gamma > 0$.
• Positivity: The solution is strictly positive in $\Omega$ and satisfies the boundary behavior $u \gtrsim \delta_\Omega$.

2. Uniqueness (Theorem 1.2)
Assuming higher regularity ($\Omega \in C^2$ and $A \in C^2(\Omega)$), the authors establish the uniqueness of distributional solutions in $L^1(\Omega)$. This result holds for a broader class of solutions than those considered in the existence theorem, as it does not a priori assume $u \in W^{1,1}_{\text{loc}}(\Omega)$.

Methodological Innovations
The paper highlights three primary technical differences from previous works on divergence-form operators (e.g., Boccardo and Orsina [6]):

• Nonlinear Gagliardo–Nirenberg Inequalities: The existence proof utilizes recently derived a priori estimates of the form $\int_\Omega \|\nabla u\|_A^2 u^{\gamma-1} \leq C \int_\Omega |Pu|u^\gamma$. These are adapted from [30] to the nondivergence setting.
• Green Function Estimates: Sharp estimates of the Green function $G_\Omega$ and its adjoint are used to derive precise decay rates for approximate solutions, which are essential for establishing the specific Sobolev regularity classes.
• Kato-type Inequalities: A novel Kato-type inequality for strong solutions in nondivergence form is established (Lemma 3.5) to prove the monotonicity of the approximating sequence. For uniqueness, a Kato-type inequality for very weak solutions is derived (Lemma 4.10), overcoming the lack of regularity in the solution space.

Significance and Scope
The authors position this work as the multidimensional counterpart to the Emden–Fowler equation for nondivergence form operators.

• Novelty: While singular semilinear equations in divergence form are well-studied, this paper addresses the case where $P$ is in nondivergence form, a class of operators central to the theory of stochastic processes and diffusion generators.
• Technical Achievement: The paper demonstrates that existence can be proven under $C^1$ coefficients for the operator, while uniqueness requires $C^2$ coefficients to handle the adjoint operator's structure and extend the class of test functions.
• Limitations and Open Problems: The authors note that the global Sobolev regularity for $\gamma \in (1, 2)$ established in Theorem 1.1(b) is not currently known for divergence-form operators. They identify open problems regarding general nonlinearities $\tau(u)$, operators with less regular coefficients, and nonlocal operators (e.g., fractional Laplacians).

The proofs rely on constructing solutions via the Schauder Fixed Point Theorem applied to regularized problems, followed by passing to the limit using the monotonicity of the sequence and the derived a priori estimates. The uniqueness proof involves extending the class of test functions to $C^{1,P}_0(\Omega)$ and utilizing weighted integrability of the source term $f/u^\gamma$ with respect to the distance function $\delta_\Omega$.