一个山路引理的应用

本文主要考虑如下形式的Dirichlet问题-△u(x)=f(x,u),x∈Ω,∈H01(Ω),其中f(x,t)∈C(Ω×R),f(x,t)/t关于t单调不减,并且当t∈R时关于x∈Ω一致趋向于某个L∞函数q(x)(此时,称f(x,t)关于t在无穷远处是渐近线性的).显然,在该条件下常用的Ambrosetti-Rabinowitz型条件,即关于所有的|s|>M和x∈Ω,0<θF(x,s)2,M>0为常数, F(x,s)=∫0s f(x,t)dt. 众所周知,条件(AR)在山路引理的应用中起着非常重要的作用.本文通过应用一种改进了的山路引理在没有条件(AR)的情况下来证明上面Dirichlet问题(P)也有正解存在。此方法也适用于f(x,t)关于t在无穷远处是超线性,即q(x)≡+∞的情形.

An Application of a Mountain Pass Theorem

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) ≡+∞.