Isolated singularities for weighted quasilinear elliptic equations |
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Authors: | Florica C Cîrstea Yihong Du |
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Institution: | a School of Mathematics and Statistics, The University of Sydney, NSW 2006, Australia b School of Science and Technology, University of New England, Armidale, NSW 2351, Australia c Department of Mathematics, Qufu Normal University, PR China |
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Abstract: | We classify all the possible asymptotic behavior at the origin for positive solutions of quasilinear elliptic equations of the form div(|∇u|p−2∇u)=b(x)h(u) in Ω?{0}, where 1<p?N and Ω is an open subset of RN with 0∈Ω. Our main result provides a sharp extension of a well-known theorem of Friedman and Véron for h(u)=uq and b(x)≡1, and a recent result of the authors for p=2 and b(x)≡1. We assume that the function h is regularly varying at ∞ with index q (that is, limt→∞h(λt)/h(t)=λq for every λ>0) and the weight function b(x) behaves near the origin as a function b0(|x|) varying regularly at zero with index θ greater than −p. This condition includes b(x)=θ|x| and some of its perturbations, for instance, b(x)=θ|x|m(−log|x|) for any m∈R. Our approach makes use of the theory of regular variation and a new perturbation method for constructing sub- and super-solutions. |
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Keywords: | Quasilinear elliptic equations Isolated singularities Regular variation theory |
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