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An Ambrosetti–Prodi-type result for a quasilinear Neumann problem

Part of: Elliptic equations and systems Variational problems in infinite-dimensional spaces

Published online by Cambridge University Press:  26 July 2012

Franciso Odair de Paiva  and
Marcelo Montenegro
Franciso Odair de Paiva
Affiliation:
Departamento de Matemática, Universidade Federal de São Carlos, 13565-905 São Carlos, SP, Brazil (odair@dm.ufscar.br)
Marcelo Montenegro
Affiliation:
IMECC, Departamento de Matemática, Universidade Estadual de Campinas, Rua Sérgio Buarque de Holanda 651, CEP 13083-859, Campinas, SP, Brazil (msm@ime.unicamp.br)
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Abstract

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We study the problem −∆pu = f(x, u) + t in Ω with Neumann boundary condition |∇u|p−2(∂u/∂v) = 0 on ∂Ω. There exists a t0 ∈ ℝ such that for t > t0 there is no solution. If t ≤ t0, there is at least a minimal solution, and for t < t0 there are at least two distinct solutions. We use the sub–supersolution method, a priori estimates and degree theory.

Keywords

a priori estimatesdegree theorysub–supersolutions

MSC classification

Secondary: 35J65: Nonlinear boundary value problems for linear elliptic equations 35J25: Boundary value problems for second-order elliptic equations 58E07: Abstract bifurcation theory
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 55 , Issue 3 , October 2012 , pp. 771 - 780
Copyright
Copyright © Edinburgh Mathematical Society 2012

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