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Paper
IPM / M / 16312 |
School of Mathematics
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Title: |
A note on the nonexistence of positive supersolutions to elliptic equations with gradient terms
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Author(s): |
Asadollah Aghajani (Joint with C. Cowan)
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Status: |
Published
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Journal: |
Annali di Matematica Pura ed Applicata
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Vol.: |
200
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Year: |
2021
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Pages: |
125-135
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Supported by: |
IPM
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Abstract: |
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 \IRN ≥ 3, then c,b must satisfy the inequality
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⎛
√
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≤ |
⎛
√
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+ |
⎛
√
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, ϕ ∈ Cc∞(Ω). |
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As an application, we obtain Liouville type theorems for positive supersolutions in exterior domains when c(x)−[(b2(x))/4] > 0 for large |x|, but unlike the known results we allow the case liminf|x|→∞c(x)−[(b2(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
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liminf
|x| →∞
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|x|2c(x) > |
(N−2+τ)2
4
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, |
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and, when τ = ∞, we have the same if
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limsup
R→∞
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R | ⎛
⎝
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inf
R < |x| < 2 R
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(c(x)− |
b(x)2
4
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) |
sup
[(R)/2] < |x| < 4 R
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|b(x)| |
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| ⎞
⎠
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=∞. |
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