Multiple positive solutions for a class of nonhomogeneous elliptic equations in esteban-lions domains with holes

Abstract. In this article, we consider the following problem -Δu + u = f(x,u) + h(x) in ⊖, u > 0 in 0, ⊖ u ∈ H₀ ¹(⊖), (*) where 0 ≤ f(x, u) ≤ aou + b _ou^p-1 for all x ∈ ⊖, u ≥ 0 with a ₀ ∈ [0, 1), 6o > 0, 2 < p < (2N/(N - 2)), if N ≥ 3, 2 < p < ∞ if N = 2 and ⊖ is the upper semi-strip domain with a hole or the upper half space with a hole. We prove that (*) has at least two positive solutions if ||h||_H-1(⊖) < C _pS(⊖)^p/2(p-2) and h ≥ 0, h ≢ E 0 in ⊖, where S(⊖) is the best Sobolev constants in S(⊖) and C_p = b₀^-1/(p-2)(p - 2)(p - 1) ^-(p-1)/(p-2)(1 - a₀)^(p-1)/(p-2).