Solvability of a Sublinear Elliptic Partial Differential Equation at Resonance

Abstract:

      This paper deals with the following elliptic partial differential equation at resonance Δｕ＋λ_１ｕ＋ｇ（ｘ，ｕ）＝０，ｘ∈Ω；ｕ｜_?Ω＝０, where Ω?Rⁿ is a bounded open set with smooth boundary and g(x,u) meets the following condition |g(x,u)|≤c|u|^α+b(x) (0≤α<1,b(x)∈L²(Ω)). We propose a condition which is not the standard Landesman Lazer condition and answer partially a question by Figueiredo and Massabò.