An Application of a Mountain Pass Theorem

We are concerned with the following Dirichlet problem: u(x)=f(x,u),x,uH10(), ((P)) where f(x, t) C (\ifmmode\expandafter\else\expandafter\=\fi), f(x, t)/t is nondecreasing in t and tends to an L -function q(x) uniformly in x as t * (i.e., f(x, t) is asymptotically linear in t at infinity). In this case, an Ambrosetti-Rabinowitz-type condition, that is, for some > 2, M > 0, 0>F(x,s)f(x,s)s,forall|s|Mandx, ((AR)) is no longer true, where F(x,s)=s0f(x,t)dt. As is well known, (AR) s 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)=+infinity.