Huan Song ZHOU
We are concerned with the following Dirichlet problem
-△u(x)=f(x,u), x∈Ω, u∈h01(Ω),(P)
where f(x,t) ∈ C(Ω×R), f(x,t)/t is nondecreasing in t ∈ R and tends to an L∞-function q(x) uniformly in x ∈Ω as t→+∞ (i.e., f(x,t) is asymptotically linear in t at infinity). In this case, Ambrosetti-Rabinowitz-type condition, that is, for someθ> 2, M > 0,
0 <θF(x,s) < f(x,s)s, for all |s| > M and x ∈Ω, (AR) is no longer true, where F(x, s) = ∫08 f(x, t)dt. As is well known, (AR) is an important technical condition in applying Mountain Pass theorem. In this paper, without assuming (AR) we prove, by using a variant version of Mountain Pass theorem, that problem (P) has a positive solution under suitable conditions on f(x,t) and q(x). Our methods also work for the case where f(x,t) is superlinear in t at infinity, i.e., q(x) ≡+∞.
