An Application of a Mountain Pass Theorem

We are concerned with the following Dirichlet problem: -△u(x) = f(x, u), x ∈Ω. u ∈ H~1(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. anAmbrosetti-Rabinowitz-type condition, that is. for some θ>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) = integral from n=0 to s f(x, t)dt. As is well known, (AR) is an important technicalcondition in applying Mountain Pass Theorem. In this paper, without assuming (AR) we prove, byusing a variant version of Mountain Pass Theorem, that problem (P) has a positive solution undersuitable, conditions on f(x, t) and q(x). Our methods also work for the case where f(x, f) is superlinearin t at infinity. i.e., q(x) ≡∞.