Positive solutions of nonlinear problems at resonance

Abstract

where L : D(L) C E -* F is a linear operator , N : E -~ F is a nonlinear operator , and E and F are Banach spaces. If L is noninvertible we say tha t (1) is a problem at resonance. In tha t case, one can use the Lyapunov-Schmidt me thod [15] or some of its extensions, such as the alternative method [1, 2, 4, 5], to give existence results for the opera tor equat ion (1). On the other hand, in numerous cases it is of interest to determine the existence of positive solutions. For instance, in the s tudy of a model of an infectious disease [7] or in the analysis of a nuclear reactor [14], By a positive solution of (1) we mean a solution which belongs to a given cone C of E . In applications, one usually takes E as a subspace of L2(A), A a domain of R n, and C as the cone of the positive functions, tha t is, C = {u ~ E : u => 0 a.e. in A}. In Section 2, we present an existence result in a cone (Theorem 2) for nonlinear problems at resonance of type (1). Theorem 2 generalizes the result in [10] since we do not require N ( C ) to be bounded. In Section 3, we use the me thod of upper and lower solutions to show the existence of positive periodic solutions for a second order nonlinear differential equat ion (Theorem 6). Taking into account the symmetries of the equation, we give sufficient conditions for the problem to have a negative periodic solution (Theorem 7). These two theorems improve some of the results of Nieto and Rao in [11].