An Application of a Mountain Pass Theorem

Abstract

We are concerned with the following Dirichlet problem: −� u(x )= f (x, u) ,x ∈ Ω ,u ∈ H 1 0 (Ω), (P) where f (x, t) ∈ C( ¯ × R), f (x, t)/t is nondecreasing in t ∈ R and tends to an L ∞ -function q(x) uniformly in x ∈ Ωa st → +∞ (i.e., f (x, t) is asymptotically linear in t at infinity). In this case,an Ambrosetti-Rabinowitz-type condition,that is,for some θ> 2 ,M> 0, 0 <θ F(x, s) ≤ f (x, s)s, for all |s |≥ M and x ∈ Ω, (AR) is no longer true,where F (x, s )= � s 0 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 )a ndq(x). Our methods also work for the case where f (x, t) is superlinear in t at infinity,i.e., q(x) ≡ +∞.