RENDICONTI DEL SEMINARIO MATEMATICO

Rend. Sem. Mat. Univ. Pol. Torino - Vol. 65, 1 (2007)Subalpine Rhapsody in DynamicsM. Franca ∗A DYNAMICAL APPROACH TO THE STUDY OF RADIALSOLUTIONS FOR P-LAPLACE EQUATIONAbstract. In this paper we give a survey of the results concerning the existence of groundstates and singular ground states for equations of the following form: p u + f (u, |x|) = 0where p u = div(|Du| p−2 Du), p > 1 is the p-Laplace operator, x ∈ R n and f is continuous,and locally Lipschitz in the u variable. We focus our attention mainly on radialsolutions.The main purpose is to illustrate a dynamical approach, which involves the introduction ofthe so called Fowler transformation. This technique turns to be particularly useful to analyzethe problem, when f is spatial dependent, critical or supercritical and to detect singularground states.1. IntroductionLet p u = div(|Du| p−2 Du), p > 1 denote the p-Laplace operator. The aim of thispaper is to discuss the existence and the asymptotic behavior of positive solutions ofequation of the following family(1) p u + f (u,|x|) = 0where p u = div(|Du| p−2 Du), p > 1, denotes the p-Laplace operator, x ∈ R n andf (u,|x|) is a continuous nonlinearity such that f (0,|x|) = 0. The interest in equationof this type started from the classical Laplacian that is p = 2:(2) u + f (u,|x|) = 0and is motivated by mathematical reasons, but also by the relevance of some equationsof this type as model to describe phenomena coming from applied area of research. Inparticular Eq. (2) is important in quantum mechanic, astronomy and chemistry, while(1) is connected to problems arising in theory of elasticity, see e.g. [26]. Our purposeis to give a short, and not exhaustive, survey of the results which can be found in thewide literature concerning this argument, and in particular to discuss a method whichis suitable to study radial solutions.We think is worthwhile to stress that Eq. (2) can be regarded as the Euler equationof the following energy functional E : R × W 1,2 (R n ) → R,∫E(x, u,∇u) =∗ The author was partially supported by G.N.A.M.P.A.(|∇u| 2− F(u,|x|) ) dx253