MULTIPLE POSITIVE SOLUTIONS FOR A CLASSOF NONHOMOGENEOUS ELLIPTIC EQUATIONSIN ESTEBAN-LIONS DOMAINS WITH HOLES

摘要

In this article, we consider the following problem —Δu + u =f(x,u) h(x)in 0Θ, u > 0 in Θ, u∈H_0~1(Θ),(*)where 0 ≤ f(x,u) ≤ a_0u b_0u~(P-1)for allx∈Θ,u ≥ 0 with a_0∈ [0, 1),b_> 2 < p (2N/(N — 2)), if N ≥ 3, 2 < p < ∞ if N = 2 and Θ is the upper semi-strip domain witha hole or the upper half space with a hole. We prove that (*) has at least two positivesolutions if < C_pS(Θ)~(P/2(P-2)) and h ≥ 0, h 0 in Θ, where S(Θ) is the best Sobolev constants in s(e) andC_p= b_0~_-1/(p-2)(p-2)(p - 1)~((p-1)/(p-2))(1-a_0)~((p-1)/(p-2)).