共振情形下次线性椭圆偏微分方程的解

Solvability of a Sublinear Elliptic Partial Differential Equation at Resonance

中文关键词:  椭圆偏微分方程  共振  Landesman-Lazer条件．

英文关键词:elliptic partial differential equation  resonance  Landesman-Lazer condition.

中文摘要:

      研究了共振下的微分方程Δｕ＋λ_１ｕ＋ｇ（ｘ，ｕ）＝０，ｘ∈?Ω；ｕ｜_?Ω＝０．在ｇ（ｘ，ｕ）关于ｕ次线性的情形，证明了解的存在性，从而部分地回答了Ｆｉｇｕｅｉｒｅｄｏ ａｎｄ Ｍａｓｓａｂｉ的一个问题．

英文摘要:

      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.