Regularizing effects for an elliptic system of singular equations

Technical Summary: Regularizing Effects for an Elliptic System of Singular Equations

Problem Statement
The paper investigates a system of two coupled, singular, semi-linear elliptic equations defined on a bounded domain $\Omega \subset \mathbb{R}^d$ ($d \geq 3$). The system is given by:
$$\begin{cases} -\text{div}(A(x)Du) + v^{1-\theta}u^{r-1} = \frac{f(x)}{u^\gamma} & \text{in } \Omega \\ -\text{div}(A(x)Dv) = \frac{u^r}{v^\theta} & \text{in } \Omega \\ u, v > 0 \text{ in } \Omega, \quad u=v=0 \text{ on } \partial\Omega \end{cases}$$
where $A(x)$ is a symmetric, uniformly elliptic matrix. The parameters satisfy $r \geq 2$, $\gamma \in (0, 1)$, and $\theta \in [0, 1)$. The datum $f$ is a non-negative function in $L^m(\Omega)$.

The system is termed "doubly singular" because both equations contain singular terms: the first equation features a singularity in the reaction term ($u^{-\gamma}$), and the second equation features a singularity in the source term ($v^{-\theta}$). This structure is modeled after the Schrödinger-Maxwell system but introduces singularities in both components. The primary objective is to establish the existence of positive solutions with finite energy and to determine how the integrability of the datum $f$ (specifically the exponent $m$) induces "regularizing effects," yielding solutions with higher integrability than typically guaranteed by the theory for single singular elliptic equations.

Methodology
The author employs an approximation scheme combined with a priori estimates, avoiding variational methods in favor of fixed-point arguments and direct energy estimates.

1. Approximation Scheme: The singular terms are regularized by truncating the datum $f$ (setting $f_n = \min\{f, n\}$) and the singular denominators (replacing $u^\gamma$ with $(u + 1/n)^\gamma$ and $v^\theta$ with $(v + 1/n)^\theta$). This yields a sequence of non-singular systems (10).
2. Existence of Approximate Solutions: For each $n$, the existence of a unique, bounded, positive weak solution $(u_n, v_n)$ is established. This is achieved by defining a map on $H^1_0(\Omega)$ and applying Schauder's fixed point theorem. The proof relies on the separate solvability of the equations with frozen coefficients and the compactness of the resulting map.
3. A Priori Estimates: The core of the analysis involves deriving uniform estimates on the sequence $(u_n, v_n)$ based on the integrability class $m$ of $f$.
   • Energy Estimates: By testing the equations with $u_n$ and $v_n$, the author establishes boundedness in $H^1_0(\Omega)$.
   • Integrability Estimates: Using test functions of the form $u_n^\delta$ and $v_n^\delta$, the author derives bounds in $L^p(\Omega)$ spaces. These bounds depend critically on the relationship between $m$, $\gamma$, $\theta$, and $r$.
4. Passage to the Limit: Using the compactness provided by the a priori estimates and Theorem 3.4 (convergence of gradients for sequences with bounded Radon measures), a subsequence converges to a limit $(u, v)$. The author carefully handles the convergence of the singular terms using Fatou's lemma, the dominated convergence theorem, and specific truncation techniques to prove that the limit satisfies the system in a weak or finite-energy sense.

Key Contributions and Results
The paper provides existence and regularity results for solutions $(u, v) \in H^1_0(\Omega) \times H^1_0(\Omega)$ across different ranges of the datum's integrability $m$:

• Case $m \geq d/2$:

  • If $m > d/2$, the solution $u$ is bounded ($u \in L^\infty(\Omega)$), and $v$ is also bounded.
  • If $m = d/2$, $u$ belongs to every $L^p(\Omega)$ for $p < \infty$, and $v \in L^\infty(\Omega)$.
  • These results recover and extend boundedness theorems known for single singular equations and non-singular Schrödinger-Maxwell systems.

• Case $m < d/2$ (Data in the Dual Space):

  • When $f$ belongs to the dual space of the first equation, specifically $(\frac{2^*}{1-\gamma})' \leq m < d/2$, the system admits a finite-energy solution.
  • The solution $u$ gains higher integrability, specifically $u \in L^{m^{**}(1+\gamma)}(\Omega)$.
  • If $r=2$, the regularity of $v$ is explicitly characterized based on $m$, $\theta$, and $\gamma$. Under specific conditions on the dimension and parameters, strict positivity of $u$ in the entire domain $\Omega$ is established.

• Case $m < d/2$ (Data Outside the Dual Space):

  • A significant contribution is the treatment of data $f$ that does not belong to the dual space of the first equation (i.e., $m < (\frac{2^*}{1-\gamma})'$).
  • Theorem 2.7: If $(\frac{r}{1-\gamma})' \leq m < (\frac{2^*}{1-\gamma})'$ and $r > 2^*$, a finite-energy solution exists with $u \in L^r(\Omega)$.
  • Theorem 2.8: If $(\frac{r+1}{1-\gamma})' \leq m < (\frac{2^*}{1-\gamma})'$, $r > 2^*-1$, and $\theta=0$, a solution exists with $u \in L^{r+1}(\Omega)$.
  • Significance: The paper notes that for single singular equations, finite-energy solutions are generally only guaranteed when the datum is in the dual space. Here, the coupling structure allows for existence even when $f$ is less regular, provided $r$ is sufficiently large.

• Higher Integrability (Theorem 2.10):

  • Under the condition $(r+\gamma)' \leq m < d/2$ and $r \geq \frac{d}{d-2} - \gamma$, the solution $u$ exhibits even higher integrability, belonging to $L^{r+1+\gamma}(\Omega)$. This demonstrates a "regularizing effect" where the singularity of the first equation, combined with the coupling, improves the integrability of the solution beyond what is standard for single equations.

Significance and Claims
The paper claims to improve upon existing results for single elliptic equations and non-singular systems. Specifically:

1. Extension of Existence Theory: It establishes the existence of finite-energy solutions for data $f$ outside the dual space of the first equation, a scenario where single singular equations typically fail to admit such solutions.
2. Regularity Gains: It quantifies how the integrability of $f$ translates into higher integrability of the solution $u$, providing explicit exponents for various ranges of $m$.
3. Doubly Singular Context: It addresses the specific challenges of a system where both equations are singular, a setting less explored than single singular equations or systems with only one singular component.
4. Methodological Rigor: The work provides a robust framework using approximation and a priori estimates to handle the lack of variational structure and the presence of singularities in both equations, ensuring the convergence of gradients and the positivity of solutions without relying on strong maximum principles that may not hold for all parameter choices.

The author emphasizes that the results are derived through rigorous a priori estimates and fixed-point arguments, avoiding assumptions on the regularity of the boundary $\partial\Omega$ beyond boundedness, relying instead on the isometric embedding of $H^1_0(\Omega)$ into $H^1(\mathbb{R}^d)$.