Bifurcation problems for the 𝑝-Laplacian in 𝑅ⁿ

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Abstract
In this paper we consider the bifurcation problem−div⁡(|∇u|p−2∇u)=λg(x)|u|p−2u+f(λ,x,u),\begin{equation*} -\operatorname {div} (|\nabla u|^{p-2}\nabla u)=\lambda g(x)|u|^{p-2}u+f(\lambda , x, u), \end{equation*}inRNR^Nwithp>1p>1. We show that a continuum of positive solutions bifurcates out from the principal eigenvalueλ1\lambda _{1}of the problem−div⁡(|∇u|p−2∇u)=λg(x)|u|p−2u.\begin{equation*}-\operatorname {div} (|\nabla u|^{p-2}\nabla u)=\lambda g(x)|u|^{p-2}u. \end{equation*}Here both functionsffandggmay change sign.

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