On quasilinear elliptic equations related to some Caffarelli-Kohn-Nirenberg inequalities

B. Abdellaoui; I. Peral

doi:10.3934/cpaa.2003.2.539

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2003, Volume 2, Issue 4: 539-566. Doi: 10.3934/cpaa.2003.2.539

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On quasilinear elliptic equations related to some Caffarelli-Kohn-Nirenberg inequalities

B. Abdellaoui^1, and
I. Peral^1,

1.
Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid

Received: December 31, 2002

Revised: June 30, 2003

Published: November 2003

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Abstract

The present work is devoted to analyze the Dirichlet problem for quasilinear elliptic equation related to some Caffarelli-Kohn-Nirenberg inequalities. Precisely the problem under study is,

-div $( |x|^{-p\gamma}|\nabla u|^{p-2}\nabla u)=f(x, u)\in L^1(\Omega),\quad x\in \Omega$

$u(x)=0$ on $\partial \Omega,$

where $-\infty<\gamma<\frac{N-p}{p}$, $\Omega$ is a bounded domain in $\mathbb R^N$ such that $0\in\Omega$ and $f(x,u)$ is a Caratheodory function under suitable conditions that will be stated in each section.

Keywords:

Degenerate and singular elliptic equations,
existence and uniqueness,
blow-up,
Cafarelli-Kohn-Nirenberg inequalities,
Weyl type lemmas.

Mathematics Subject Classification: 35D05, 35D10, 35J20, 35J25, 35J70.

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