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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Weighted Sobolev type embeddings and coercive quasilinear elliptic equations on $ \mathbb{R}^N$


Authors: Jiabao Su and Rushun Tian
Journal: Proc. Amer. Math. Soc. 140 (2012), 891-903
MSC (2010): Primary 35J05, 35J20, 35J60, 58C20
Published electronically: August 15, 2011
MathSciNet review: 2869073
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Abstract: We study weighted Sobolev type embeddings of radially symmetric functions from $ W_r^{1,p}(\mathbb{R}^N; V)$ into $ L^q(\mathbb{R}^N; Q)$ for $ q<p$ with singular potentials. We then investigate the existence of nontrivial radial solutions of quasilinear elliptic equations with singular potentials and sub-$ p$-linear nonlinearity. The model equation is of the form

\begin{displaymath}\begin{cases}-\hbox{div}(\vert\nabla u\vert^{p-2}\nabla u)+V... ...x)\rightarrow0, \quad \vert x\vert\rightarrow\infty.\end{cases}\end{displaymath}


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Additional Information

Jiabao Su
Affiliation: School of Mathematical Sciences, Capital Normal University, Beijing 100048, People’s Republic of China

Rushun Tian
Affiliation: Department of Mathematics and Statistics, Utah State University, Logan, Utah 84322

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11289-9
PII: S 0002-9939(2011)11289-9
Keywords: Quasilinear elliptic equation, Sobolev type embedding, sub-$p$-linear nonlinearity
Received by editor(s): December 13, 2010
Published electronically: August 15, 2011
Additional Notes: This work was supported by NSFC-10831005, PHR201106118, and KZ201010028027
Communicated by: Walter Craig
Article copyright: © Copyright 2011 American Mathematical Society