Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Singular elliptic equations having a gradient term with natural growth

A. Ferone, A. Mercaldo, S. Segura de León

Abstract

We study a class of Dirichlet boundary value problems whose prototype is \begin{equation}\label{1.2abs} \left\{\begin{array}{ll} -Δ_p u =h(u)|\nabla u|^p+u^{q-1}+f(x)\, &\quad\hbox{in } \ Ω\,,\\ u\ge 0\,,&{\quad\hbox{in } \ Ω}\\ u = 0\,&\quad\hbox{on }\partial Ω\,,\end{array}\right. \end{equation} where $Ω$ an open bounded subset of $\mathbb R^N$, $0<q<1$, $1<p<N$, $h$ is a continuous function and $f$ belongs to a suitable Lebesgue space. The main features of this problem are the presence of a singular term and a first order term with natural growth in the gradient. A priori estimates and existence results are proved depending on the summability of the datum $f$.

Singular elliptic equations having a gradient term with natural growth

Abstract

We study a class of Dirichlet boundary value problems whose prototype is \begin{equation}\label{1.2abs} \left\{\begin{array}{ll} -Δ_p u =h(u)|\nabla u|^p+u^{q-1}+f(x)\, &\quad\hbox{in } \ Ω\,,\\ u\ge 0\,,&{\quad\hbox{in } \ Ω}\\ u = 0\,&\quad\hbox{on }\partial Ω\,,\end{array}\right. \end{equation} where an open bounded subset of , , , is a continuous function and belongs to a suitable Lebesgue space. The main features of this problem are the presence of a singular term and a first order term with natural growth in the gradient. A priori estimates and existence results are proved depending on the summability of the datum .
Paper Structure (16 sections, 9 theorems, 160 equations)

This paper contains 16 sections, 9 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Notation, assumptions and preliminary results
  3. Assumptions
  4. Auxiliary functions
  5. Approximating problems
  6. Existence result for $m> N/p$
  7. A priori estimates when $m>N/p$
  8. Strong convergence of $\nabla u_n$
  9. Existence: proof of Theorem \ref{['main']}
  10. Existence result for $m= N/p$
  11. A priori estimates when $m=N/p$
  12. Strong convergence of $\nabla u_n$
  13. Existence: proof of Theorem \ref{['mainlimit']}
  14. Existence result for $\frac{Np}{Np-N+p}\le m<\frac{N}{p}$
  15. A priori estimates
  16. ...and 1 more sections

Key Result

Lemma 2.1

(Cancellation lemma) Let $u_n \in W^{1, p}_0 ( \Omega) \cap L^{\infty} ( \Omega )$ be a weak solution to problem (pd1).

Theorems & Definitions (18)

  • Remark 2.1
  • Definition 2.1
  • Lemma 2.1
  • Theorem 3.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 4.1
  • Lemma 4.1
  • ...and 8 more