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Paper   IPM / M / 16727
School of Mathematics
  Title:   Positive supersolutions of fourth-order nonlinear elliptic equations: explicit estimates and Liouville theorems
  Author(s):  Asadollah Aghajani (Joint with C. Cowan and V. D. Radulescu)
  Status:   Published
  Journal: Journal of Differential Equations
  Vol.:  298
  Year:  2021
  Pages:   323-345
  Supported by:  IPM
  Abstract:
In this paper, we consider positive supersolutions of the semilinear fourth-order problem


(−∆)2 u=ρ(x) f(u)
in  Ω, ∆u > 0
in Ω,
where Ω is a domain in \IRN (bounded or not), f:Df = [0,af) → [0,∞) (0 < af \leqslant +∞) is a non-decreasing continuous function with f(u) > 0 for u > 0 and ρ: Ω→ \IR 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 supx ∈ Ωdist (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 ∆(√{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 −∆u ≥ √{2ρ(x)F(u)} for all x ∈ Ω, where F(t):=∫0t (f(s)−f(0))ds.

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