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Proceedings of the American Mathematical Society

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On eigenvalue problems of $ p$-Laplacian with Neumann boundary conditions


Author: Yin Xi Huang
Journal: Proc. Amer. Math. Soc. 109 (1990), 177-184
MSC: Primary 35P15; Secondary 35J65
DOI: https://doi.org/10.1090/S0002-9939-1990-1010800-9
MathSciNet review: 1010800
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Abstract: We study the nonlinear eigenvalue problem

$\displaystyle - {\Delta _p}u = \lambda m(x)\vert u{\vert^{p - 2}}u\,\quad {\tex... ...text{on}}\,\partial \Omega ,\,\;{\text{where}}\,p > 1,\lambda \in {\mathbf{R}}.$

For $ \int_\Omega {m(x) < 0} $, we prove that the first positive eigenvalue $ {\lambda _1}$ exists and is simple and unique, in the sense that it is the only eigenvalue with a positive eigenfunction. In the case $ \int_\Omega {m(x) = 0} $, we prove that $ {\lambda _0} = 0$ is the only eigenvalue with a positive eigenfunction.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1990-1010800-9
Keywords: $ p$-Laplacian, eigenvalue, Neumann boundary condition
Article copyright: © Copyright 1990 American Mathematical Society

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