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Transactions of the American Mathematical Society

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Bifurcation problems for the $p$-Laplacian in $R^N$

Authors: Pavel Drábek and Yin Xi Huang
Journal: Trans. Amer. Math. Soc. 349 (1997), 171-188
MSC (1991): Primary 35B32, 35J70, 35P30
MathSciNet review: 1390979
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Abstract: In this paper we consider the bifurcation problem

\begin{equation*}-\text {div\ } (|{\nabla } u|^{p-2}{\nabla } u)={\lambda } g(x)|u|^{p-2}u+f({\lambda } , x, u), \end{equation*}

in ${R^N} $ with $p>1$. We show that a continuum of positive solutions bifurcates out from the principal eigenvalue ${\lambda } _{1}$ of the problem

\begin{equation*}-\text {div\ } (|{\nabla } u|^{p-2}{\nabla } u)={\lambda } g(x)|u|^{p-2}u. \end{equation*}

Here both functions $f$ and $g$ may change sign.

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Additional Information

Pavel Drábek
Affiliation: Department of Mathematics, University of West Bohemia, P.O. Box 314, 30614 Pilsen, Czech Republic

Yin Xi Huang
Affiliation: Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee 38152

Keywords: $p$-Laplacian, global positive solutions, weighted spaces
Received by editor(s): November 18, 1994
Received by editor(s) in revised form: March 10, 1995
Additional Notes: The first author was partially supported by the Grant Agency of the Czech Republic under the Grant No. 201/94/0008
Article copyright: © Copyright 1997 American Mathematical Society