Fowler transformation and radial solutions for quasilinear elliptic equations. Part 2: nonlinearities of mixed type

Abstract

We discuss the existence and the asymptotic behavior of positive radial solutions for the following equation: $$ \Delta_p u({\bf x})+f(u,|{\bf x}|)=0, $$ where \({\Delta_p u={\rm div}(|Du|^{p-2}Du), {\bf x} \in \mathbb{R}^n, n > p > 1}\) , and we assume that f ≥ 0 is subcritical for u large and |x| small and supercritical for u small and |x| large, with respect to the Sobolev critical exponent. We give sufficient conditions for the existence of ground states with fast decay. As a corollary we also prove the existence of ground states with slow decay and of singular ground states with fast and slow decays. For the proofs we use a Fowler transformation which enables us to use dynamical arguments. This approach allows to unify the study of different types of non-linearities and to complete the results already appeared in literature with the analysis of singular solutions.