The sublinear problem for the 1-homogeneous 𝑝-Laplacian

Abstract

In this paper we prove the existence and uniqueness of a positive viscosity solution of the 1 1 -homogeneous p p -Laplacian with a sublinear right-hand side; that is, − | D u | 2 − p div ( | D u | p − 2 D u ) = λ u q -|D u|^{2-p}\textrm {div}\:(|D u|^{p-2}Du)=\lambda u^q in Ω \Omega , u = 0 u=0 on ∂ Ω \partial \Omega , where Ω \Omega is a bounded starshaped domain, λ > 0 \lambda >0 , p > 2 p>2 and 0 > q > 1 0>q>1 .