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Let \( \Omega \subset \mathbb {R}^N \) , with \( N > 1 \) , be a bounded domain. In this work, we investigate the existence of a renormalized solution to the following nonlinear elliptic system: 1 \(\begin{aligned} \left\{ \begin{array}{ll} -\operatorname {div}(A_{1}(x,u_{2})\, Du_{1}) = f & \text {in } \Omega , \\ -\operatorname {div}(A_{2}(x,u_{2})\, Du_{2}) + H(x,u_{2})\, |Du_{2}|^{2} = A_{1}(x,u_{2})\, Du_{1} \cdot Du_{1} & \text {in } \Omega , \\ u_1 = 0, \quad u_2 = 0 & \text {on } \partial \Omega . \end{array} \right. \end{aligned}\) A diffusion matrix \( A_2 \) may blow up when \( u_2 \) approaches a finite critical value \( m > 0 \) . The matrix \( A_1 \) is assumed to be bounded and symmetric. The source term satisfies \( f \in L^{\infty }(\Omega )\) , and the absorption coefficient \( H : \Omega \times (m, +\infty ) \rightarrow \mathbb {R}^{+} \) is a Carathéodory function.

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Analysis of Elliptic Systems with Singular Diffusion Coefficients

  • 会议论文

  • 发表日期：

    2026年

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  • Mohammed El Fatry,
  • Mounir Mekkour,
  • Youssef Akdim

摘要

Let \( \Omega \subset \mathbb {R}^N \) , with \( N > 1 \) , be a bounded domain. In this work, we investigate the existence of a renormalized solution to the following nonlinear elliptic system: 1 \(\begin{aligned} \left\{ \begin{array}{ll} -\operatorname {div}(A_{1}(x,u_{2})\, Du_{1}) = f & \text {in } \Omega , \\ -\operatorname {div}(A_{2}(x,u_{2})\, Du_{2}) + H(x,u_{2})\, |Du_{2}|^{2} = A_{1}(x,u_{2})\, Du_{1} \cdot Du_{1} & \text {in } \Omega , \\ u_1 = 0, \quad u_2 = 0 & \text {on } \partial \Omega . \end{array} \right. \end{aligned}\) A diffusion matrix \( A_2 \) may blow up when \( u_2 \) approaches a finite critical value \( m > 0 \) . The matrix \( A_1 \) is assumed to be bounded and symmetric. The source term satisfies \( f \in L^{\infty }(\Omega )\) , and the absorption coefficient \( H : \Omega \times (m, +\infty ) \rightarrow \mathbb {R}^{+} \) is a Carathéodory function.