Bifurcation problems for the 𝑝-Laplacian in 𝑅ⁿ

Abstract

In this paper we consider the bifurcation problem − div ⁡ ( | ∇ u | p − 2 ∇ u ) = λ g ( x ) | u | p − 2 u + 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*} in R N R^N with p > 1 p>1 . We show that a continuum of positive solutions bifurcates out from the principal eigenvalue λ 1 \lambda _{1} of the problem − div ⁡ ( | ∇ u | p − 2 ∇ u ) = λ g ( x ) | u | p − 2 u . \begin{equation*}-\operatorname {div} (|\nabla u|^{p-2}\nabla u)=\lambda g(x)|u|^{p-2}u. \end{equation*} Here both functions f f and g g may change sign.