Bifurcation problems for the 𝑝-Laplacian in 𝑅ⁿ

Drábek, Pavel; Huang, Yin

doi:10.1090/S0002-9947-97-01788-1

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
DOI: https://doi.org/10.1090/S0002-9947-97-01788-1
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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