Advances in Differential Equations
Volume 9, Numbers 9-10, Pages 1185–1200
September/October 2004
ANTIMAXIMUM PRINCIPLE FOR QUASILINEAR
PROBLEMS
David Arcoya
Departamento de Análisis Matemático, Universidad de Granada
Granada 18071, Spain
Julio D. Rossi
Departamento de Matemática, Universidad Católica de Chile
Casilla 306, Correo 22, Santiago, Chile
(Submitted by: Jean Mawhin)
Abstract. In this paper, for a bounded, smooth domain Ω ⊂ RN ,
h ∈ Lr (Ω), r > N/2, g ∈ Ls (∂Ω), s > N − 1, we prove the maximum
and antimaximum principle for the quasilinear boundary-value problem
−div(A(u)∇u) + u = h in Ω, A(u)∇u · η = λf (u) + g, on ∂Ω, where A
is elliptic and bounded and f is asymptotically linear. The sharpness of
this result (r > N/2 and s > N −1) is discussed for the linear boundaryvalue problem, −∆u + u = h in Ω, ∂u/∂η = λu + g on ∂Ω.
1. Introduction
In [7] the well-known maximum/antimaximum principle for the linear
Laplace operator with zero Dirichlet boundary condition is proved. Denoting by µ1 the first eigenvalue of the Laplace operator (with zero Dirichlet
boundary condition), it states that “given any positive h ∈ Lr (Ω), r > N ,
there exists ε = ε(h) > 0 such that every solution u of −∆u = µu + h in
Ω with u |∂Ω = 0 satisfies either u < 0 in Ω provided that µ1 < µ < µ1 + ε
(antimaximum principle), or u > 0 in Ω if µ < µ1 (maximum principle).”
The main goal of this paper is to put in evidence the nonlinear nature
of this principle. Indeed, we extend the principle to the case of elliptic
quasilinear equations with a nonlinear boundary condition. Since we also
want to point out some differences between the case of a Dirichlet boundary
condition and a Neumann one, we study here the associated boundary-value
problem with a nonlinear Neumann boundary condition. Specifically, we
Accepted for publication: March 2004.
AMS Subject Classifications: 35P05, 35J60, 35J55.
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