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Regularizing effect of absorption terms in singular and degenerate elliptic problems

Abdelaaziz Sbai, Youssef El Hadfi

Abstract

In this paper we study the existence and regularity of solutions to the following singular problem \begin{equation} \left\{ \begin{array}{lll} &-\displaystyle\mbox{div} \big(a(x,u)|\nabla u|^{p-2}|\nabla u|\big) + |u|^{s-1}u =\frac{f}{u^γ} &\mbox{ in } Ω\\ &u>0 &\mbox{ in }Ω\\ &u=0 &\mbox{ on } δΩ\end{array} \right. \end{equation} proving that the lower order term $u|u|^{s-1}$ has some regularizing effects on the solutions in the case of an elliptic operator with degenerate coercivity.

Regularizing effect of absorption terms in singular and degenerate elliptic problems

Abstract

In this paper we study the existence and regularity of solutions to the following singular problem \begin{equation} \left\{ \begin{array}{lll} &-\displaystyle\mbox{div} \big(a(x,u)|\nabla u|^{p-2}|\nabla u|\big) + |u|^{s-1}u =\frac{f}{u^γ} &\mbox{ in } Ω\\ &u>0 &\mbox{ in }Ω\\ &u=0 &\mbox{ on } δΩ\end{array} \right. \end{equation} proving that the lower order term has some regularizing effects on the solutions in the case of an elliptic operator with degenerate coercivity.

Paper Structure

This paper contains 6 sections, 8 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. Statement of definitions and the main results
  3. Statement of definitions
  4. Statement of the main results
  5. A priori estimates and Preliminary facts
  6. Proof of the results

Key Result

Theorem 2.1

Let $f\in L^{m}(\Omega)$, $m>1$, $1<p<N$. Then

Theorems & Definitions (16)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.1
  • Remark 2.1
  • Theorem 2.2
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.3
  • Theorem 2.4
  • ...and 6 more