Differential and Integral Equations

Quasilinear elliptic equations with natural growth

Boumediene Abdellaoui, Lucio Boccardo, Ireneo Peral, and Ana Primo

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In this paper we deal with the problem $$\left\{ \begin{array}{rcl} - {\rm div}\, (a(x,u)\nabla u) +{g(x,u,\nabla u)} & = & \lambda h(x)u + f{\mbox{ in }}\Omega,\\ u & = & 0{\mbox{ on }}\partial\Omega. \end{array} \right. $$ The main goal of the work is to get hypotheses on $a$, $g$ and $h$ such that the previous problem has a solution for all $\lambda>0$ and $f\in L^1(\Omega)$. In particular, we focus our attention in the model equation with $a(x,u)= (1+|u|^m)$, $g(x,u,\nabla u)=\frac{m}{2}|u|^{m-2}u|\nabla u|^2$ and $h(x)=\dfrac{1}{|x|^2}$.

Article information

Differential Integral Equations, Volume 20, Number 9 (2007), 1005-1020.

First available in Project Euclid: 20 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J60: Nonlinear elliptic equations
Secondary: 35D05 35D10 35J20: Variational methods for second-order elliptic equations 35J70: Degenerate elliptic equations 47J30: Variational methods [See also 58Exx]


Abdellaoui, Boumediene; Boccardo, Lucio; Peral, Ireneo; Primo, Ana. Quasilinear elliptic equations with natural growth. Differential Integral Equations 20 (2007), no. 9, 1005--1020. https://projecteuclid.org/euclid.die/1356039308

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