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Positive supersolutions of fourth-order nonlinear elliptic equations: explicit estimates and Liouville theorems
Journal of Differential Equations ( IF 2.4 ) Pub Date : 2021-07-15 , DOI: 10.1016/j.jde.2021.07.005
Asadollah Aghajani , Craig Cowan , Vicenţiu D. Rădulescu

In this paper, we consider positive supersolutions of the semilinear fourth-order problem{(Δ)2u=ρ(x)f(u)inΩ,Δu>0inΩ, where Ω is a domain in RN (bounded or not), f:Df=[0,af)[0,) (0<af+) is a non-decreasing continuous function with f(u)>0 for u>0 and ρ:ΩR is a positive function. Using a maximum principle-based argument, we give explicit estimates on positive supersolutions that can easily be applied to obtain Liouville-type results for positive supersolutions either in exterior domains, or in unbounded domains Ω with the property that supxΩdist(x,Ω)=. In particular, we consider the above problem with f(u)=up (p>0) and with different weights ρ(x)=|x|a, eax1 or x1m (m is an even integer). Also, when f is convex and ρ:Ω(0,) is smooth with Δ(ρ)>0, then under an extra condition between f and ρ we show that every positive supersolution u of this problem with u=0 on ∂Ω (Ω bounded) satisfies the inequality Δu2ρ(x)F(u) for all xΩ, where F(t):=0t(f(s)f(0))ds.



中文翻译:

四阶非线性椭圆方程的正超解:显式估计和刘维尔定理

在本文中,我们考虑半线性四阶问题的正超解{(-Δ)2=ρ(X)F()Ω,-Δ>0Ω, 其中 Ω 是一个域 电阻N (有界或无界), FDF=[0,一种F)[0,) (0<一种F+) 是一个非递减的连续函数 F()>0 为了 >0ρΩ电阻是正函数。使用基于最大原理的论证,我们给出了对正超解的明确估计,这些估计可以很容易地应用于在外部域或无界域 Ω 中获得正超解的 Liouville 型结果,其性质为XΩ(X,Ω)=. 特别地,我们考虑上述问题F()= (>0) 和不同的权重 ρ(X)=|X|一种, 电子一种X1 或者 X1m是偶数)。此外,当f是凸的并且ρΩ(0,) 是光滑的 Δ(ρ)>0,然后在fρ之间的一个额外条件下,我们证明了这个问题的每个正超解u=0 on ∂Ω (Ω bounded) 满足不等式 -Δ2ρ(X)F() 对所有人 XΩ, 在哪里 F()=0(F()-F(0))d.

更新日期:2021-07-15
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