On Multiple Positive Solutions of Positone and Non-positone Problems

In this paper, we consider the following problem: − u= f (u) in , u= 0 on ∂ , (1.1) where is the ball BR = {x ∈ R ; |x| < R}, | · | is the Euclidean norm in R, and f : R+ → R is a locally Lipschitzian continuous function. We are concerned with two classes of problems, namely, (i) the positone problem: f (0)≥ 0; (ii) the non-positone problem: f (0) < 0. The study of positone problems was initiated by Keller and Cohen [14], see also [15], motivated by problems arose from the theory of nonlinear heat generation. In the past twenty five years there has been considerable interest in this class of semilinear elliptic boundary value problems and there is a wide literature on this subject. The reader may consult the survey by Lions [17], and the references therein, where many interesting questions are studied under a different point of view. In the positone case we consider the following assumptions: (f1) there exist 0 < a1 < a2 < a3 so that f (a1) = f (a2) = 0 and F(a3) > F(a1), where F(t)= ∫ t 0 f ; (f2) |f (t)| ≤ α, for all t ∈ R+, for some constant α > 0; (P) f (0) > 0 or, if f (0) = 0, then f ′ +(0) > 0, where f ′ +(0) = limt→0+(f (t)/t) and prove the result below.