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

 

Existence and nonexistence of positive eigenfunctions for the $ p$-Laplacian


Authors: P. A. Binding and Y. X. Huang
Journal: Proc. Amer. Math. Soc. 123 (1995), 1833-1838
MSC: Primary 35P30; Secondary 47H12, 47N20
MathSciNet review: 1260160
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Abstract: We study the relation between (i) the principal eigencurve, i.e., the graph of $ {\mu _1}$ satisfying

$\displaystyle - {\operatorname{div}}(\vert\nabla u{\vert^{p - 2}}\nabla u) + (q... ...mbda w(x))\vert u{\vert^{p - 2}}u = {\mu _1}(\lambda )\vert u{\vert^{p - 2}}\mu$ ($ \ast $)

on a smooth bounded domain $ \Omega $ in $ {{\mathbf{R}}^N}$ with $ p > 1$, and (ii) existence and nonexistence of positive solutions of $ ( \ast )$ with $ {\mu _1}(\lambda ) = 0$. Eigencurve arguments are used extensively.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1260160-8
PII: S 0002-9939(1995)1260160-8
Article copyright: © Copyright 1995 American Mathematical Society