Positive supersolutions of fourth-order nonlinear elliptic equations: explicit estimates and Liouville theorems

Abstract

In this paper, we consider positive supersolutions of the semilinear fourth-order problem { ( − Δ ) 2 u = ρ ( x ) f ( u ) in Ω , − Δ u > 0 in Ω , where Ω is a domain in R N (bounded or not), f : D f = [ 0 , a f ) → [ 0 , ∞ ) ( 0 < a f ⩽ + ∞ ) is a non-decreasing continuous function with f ( u ) > 0 for u > 0 and ρ : Ω → R 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 sup x ∈ Ω ⁡ dist ( x , ∂ Ω ) = ∞ . In particular, we consider the above problem with f ( u ) = u p ( p > 0 ) and with different weights ρ ( x ) = | x | a , e a x 1 or x 1 m ( 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 − Δ u ≥ 2 ρ ( x ) F ( u ) for all x ∈ Ω , where F ( t ) : = ∫ 0 t ( f ( s ) − f ( 0 ) ) d s .