Multiple positive solutions for an inhomogeneous semilinear problem in exterior domains Original Research Article

This paper is a contribution on the inhomogeneous problem View the MathML source{Δu+K(x)up+λf(x)=0in Ω,u>0in Ω,u∈Hloc1(Ω)∩C(Ω¯),u|∂Ω=0,u→μ>0as |x|→∞, Turn MathJax on where Ω=RN∖ωΩ=RN∖ω is an exterior domain in RNRN, ω⊂RNω⊂RN is a bounded domain with a smooth boundary and N>2N>2. λ>0λ>0, μ>0μ>0 and p>1p>1 are given constants. f(x)∈L∞(Ω)f(x)∈L∞(Ω) and K(x)K(x) are given locally Hölder continuous functions in View the MathML sourceΩ̄, and K(x)K(x) satisfies a fast decay condition: ∃C,ϵ,M>0∃C,ϵ,M>0 such that |K(x)|≤C|x|l|K(x)|≤C|x|l for any |x|≥M|x|≥M with l≤−2−ϵl≤−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)K(x) and f(x)f(x).