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2006 Positive solutions to singular semilinear elliptic equations with critical potential on cone-like domains
Vitali Liskevich, Sofya Lyakhova, Vitaly Moroz
Adv. Differential Equations 11(4): 361-398 (2006).

Abstract

We study the existence and nonexistence of positive (super-) solutions to a singular semilinear elliptic equation $$-\nabla\cdot(|x|^A\nabla u)-B|x|^{A-2}u=C|x|^{A-\sigma}u^p$$ in cone--like domains of $\mathbb R^N$ ($N\ge 2$), for the full range of parameters $A,B,\sigma,p\in\mathbb R$ and $C>0$. We provide a characterization of the set of $(p,\sigma)\in\mathbb R^2$ such that the equation has no positive (super-),solutions, depending on the values of $A,B$ and the principal Dirichlet eigenvalue of the cross--section of the cone. The proofs are based on the explicit construction of appropriate barriers and involve the analysis of asymptotic behavior of super-harmonic functions associated to the Laplace operator with critical potentials, Phragmén-Lindelöf type comparison arguments and an improved version of Hardy's inequality in cone--like domains.

Citation

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Vitali Liskevich. Sofya Lyakhova. Vitaly Moroz. "Positive solutions to singular semilinear elliptic equations with critical potential on cone-like domains." Adv. Differential Equations 11 (4) 361 - 398, 2006.

Information

Published: 2006
First available in Project Euclid: 18 December 2012

zbMATH: 1194.35170
MathSciNet: MR2215620

Subjects:
Primary: 35J60
Secondary: 35B05, 35B33

Rights: Copyright © 2006 Khayyam Publishing, Inc.

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Vol.11 • No. 4 • 2006
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