Multiple positive solutions for an inhomogeneous semilinear problem in exterior domains

This paper is a contribution on the inhomogeneous problem {Δu+K(x)up+λf(x)=0in Ω,u>0in Ω,u∈Hloc1(Ω)∩C(Ω¯),u|∂Ω=0,u→μ>0as |x|→∞, where Ω=RN∖ω is an exterior domain in RN, ω⊂RN is a bounded domain with a smooth boundary and N>2. λ>0, μ>0 and p>1 are given constants. f(x)∈L∞(Ω) and K(x) are given locally Hölder continuous functions in Ω̄, and K(x) satisfies a fast decay condition: ∃C,ϵ,M>0 such that |K(x)|≤C|x|l for any |x|≥M with l≤−2−ϵ. By applying the monotone iteration method and the Mountain Pass Lemma, some results on the existence and nonexistence of multiple solutions are discussed under different assumptions for K(x) and f(x).