Quasilinear equations with a multiple bifurcation
type: Item
instance of: scholarly article
title: Quasilinear equations with a multiple bifurcation
zbMATH Open document ID: 0879.35021
published in: Differential and Integral Equations
publication date: 1997-12-10T00:00:00Z
review text: Radial solutions of quasilinear boundary value problems of the type \[ -\Delta_p u = \lambda h(u) + g(r,u)\;\text{ for } r=|x|<1,\;u=0\;\text{ for } r=|x|=1 \tag \(\ast\) \] are studied where \(\Delta_p\) is the \(p\)-Laplace operator in \(\mathbb{R}^N,\;\lambda \geq 0,\;h(u) \simeq |u|^{q-2}u,\;1<q<p\) near \(u=0\), \(g\) is of higher order with respect to \(h\) at \(u=0\). The existence of infinitely many different continua of radial solutions of (\(\ast \)) emanating from \([\lambda ,u]=[0,0]\) is proved. Each continuum is characterized by nodal properties. As a consequence, the existence of infinitely many radial solutions follows for any \(\lambda >0\) small enough. The nonlinearities are not supposed to be odd and no growth restrictions are considered. Global behaviour of the bifurcating branches is described under different assumptions on \(g\). Particularly, if \(g(u) \simeq |u|^{\alpha -2}u\) as \(|u|\to \infty \), \(\alpha < p^*\) (the subcritical growth) then each continuum mentioned is bounded for \(\lambda \geq 0\) in the \(L_\infty\) norm and (*) possesses infinitely many pairs of radial solutions for any \(\lambda \geq 0\) small enough. Further, the case \(h(u)=u^{q-1},\;g(u)=u^{\alpha -1}\) for \(u \geq 0\) with \(\alpha \leq p^*\) is considered. In this case the set of all positive radial solutions forms a branch. This branch blows up to infinity as \(\lambda \to 0_+\) if \(\alpha =p^*\) and \(p,q\) satisfy certain additional assumptions.
Mathematics Subject Classification ID: 35B32
Mathematics Subject Classification ID: 35J65
zbMATH DE Number: 1095820
zbMATH Keywords: global bifurcation
zbMATH Keywords: quasilinear equations
zbMATH Keywords: multiplicity
zbMATH Keywords: nodal properties
author: A. Ambrosetti
author: Jesús Garcia-Azorero
author: Ireneo Peral
reviewed by: Milan Kučera
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1095820