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Existence of positive solutions for some problems with nonlinear diffusion

Authors: A. Cañada, P. Drábek and J. L. Gámez
Journal: Trans. Amer. Math. Soc. 349 (1997), 4231-4249
MSC (1991): Primary 35J65, 35J55; Secondary 47H17, 58E30, 92D25
MathSciNet review: 1422596
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Abstract: In this paper we study the existence of positive solutions for problems of the type

\begin{equation*}\begin {array}{cl} -\Delta _pu(x)=u(x)^{q-1}h(x,u(x)), & x\in \Omega , \\ u(x)=0, & x\in \partial \Omega , \end {array} \end{equation*}

where $\Delta _p$ is the $p$-Laplace operator and $p,q>1$. If $p=2$, such problems arise in population dynamics. Making use of different methods (sub- and super-solutions and a variational approach), we treat the cases $p=q$, $p<q$ and $p>q$, respectively. Also, some systems of equations are considered.

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Additional Information

A. Cañada
Affiliation: Department of Mathematical Analysis, University of Granada, 18071, Granada, Spain

P. Drábek
Affiliation: Department of Mathematics, University of West Bohemia Plzen, Americká 42, 306 14 Plzen, Czech Republic

J. L. Gámez
Affiliation: Department of Mathematical Analysis, University of Granada, 18071, Granada, Spain

Keywords: Nonlinear elliptic problems, boundary value problems, positive solutions, nonlinear diffusion, sub- and super-solutions, variational methods
Received by editor(s): October 11, 1994
Received by editor(s) in revised form: May 6, 1996
Additional Notes: The first and the third author have been supported in part by DGICYT, Ministry of Education and Science (Spain), under grant number PB95-1190 and by EEC contract, Human Capital and Mobility program, ERBCHRXCT940494. The second author was partially supported by the Grant Agency of the Czech Republic under Grant No. 201/94/0008, and he is grateful to University of Granada for pleasant hospitality during preparation of this paper.
Article copyright: © Copyright 1997 American Mathematical Society

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