An Application of a Mountain Pass Theorem

摘要

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. anAmbrosetti-Rabinowitz-type condition, that is. for some θ2. M0, 0θF（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） ≡∞.