Electron. J. Diff. Eqns., Vol. 1995(1995), No. 10, pp 1-22.

Existence of Positive Solutions for some Dirichlet Problems with an Asymptotically Homogeneous Operator

Marta Garcia-Huidobro, Raul Manasevich, & Pedro Ubilla

Abstract:
Existence of positive radially symmetric solutions to a Dirichlet problem of the form
$$\eqalign {{\rm div\,} (A(|Du|)Du)&=f(u)\quad { \rm in\ } \Omega \cr u &= 0 \quad { \rm on\ } \partial\Omega} $$
is studied by using blow-up techniques. It is proven here that by choosing the functions $sA(s)$ and $f(s)$ among a certain class called asymptotically homogeneous, the blow-up method still provides the a-priori bounds for positive solutions. Existence is proved then by using degree theory.

Submitted February 12, 1995. Published August 11, 1995.
Math Subject Classification: 35J65.
Key Words: Dirichlet Problem, Positive Solution, Blow up.

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Marta Garcia-Huidobro
Departamento de Matematicas
Facultad de Matematica
Universidad Catolica de Chile
Casilla 306, Correo 22, Santiago, Chile
e-mail: mgarcia@poincare.mat.puc.cl
Raul Manasevich
Departamento de Ingenieria Matematica, F.C.F.M., Universidad de Chile
Casilla 170, Correo 3, Santiago, Chile
e-mail: manasevi@dim.uchile.cl
Pedro Ubilla
Departamento de Matematicas, Universidad de Santiago de Chile
Casilla 307, Correo 2, Santiago, Chile
e-mail: pubilla@fermat.usach.cl

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