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

Abstract:

In this paper we consider the bifurcation problem \begin{equation*} -\operatorname {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*}-\operatorname {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.