Existence of multiple positive solutions for inhomogeneous Neumann problem

In this paper, we study the existence and nonexistence of multiple positive solutions for the inhomogeneous Neumann boundary value problem Deltau + u(P) - lambdau = 0, with Dgammau = phi(x), under some assumptions on the boundary partial derivativeOmega and the function phi(x). For phi(x) greater than or equal to 0, phi(x) not equivalent to 0, phi(x) is an element of C-alpha((Ω) over bar), it is shown that there exists a constant lambda* > 0 such that problem (*) possesses at least two positive solutions if lambda is an element of (lambda*, infinity) and at least one positive solution if lambda = lambda*. Furthermore, there are no positive solutions for problem (*) if lambda is an element of (-infinity, lambda*). (C) 2002 Elsevier Science (USA). All rights reserved.