We prove that if the elliptic problem −∆u+b(x)∇u=c(x)u with c ≥ 0 has a positive supersolution in a domain Ω of \IR^{N ≥ 3}, then c,b must satisfy the inequality

⎛ √


≤ 
⎛ √


+ 
⎛ √


, ϕ ∈ C_{c}^{∞}(Ω). 

As an application, we obtain Liouville type theorems for positive supersolutions in exterior domains when c(x)−[(b^{2}(x))/4] > 0 for large x, but unlike the known results we allow the case liminf_{x→∞}c(x)−[(b^{2}(x))/4]=0. Also the weights b and c are allowed to be unbounded. In particular, among other things, we show that if τ:=limsup_{x →∞}xb(x) < ∞ then this problem does not admit any positive supersolution if

liminf
x →∞

x^{2}c(x) > 
(N−2+τ)^{2}
4

, 

and, when τ = ∞, we have the same if

limsup
R→∞

R  ⎛ ⎝



inf
R < x < 2 R

(c(x)− 
b(x)^{2}
4

) 
sup
[(R)/2] < x < 4 R

b(x) 

 ⎞ ⎠

=∞. 

Download TeX format
