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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Estimates for eigenfunctions and eigenvalues of nonlinear elliptic problems

Author: Chris Cosner
Journal: Trans. Amer. Math. Soc. 282 (1984), 59-75
MSC: Primary 35P30; Secondary 35J55
MathSciNet review: 728703
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Abstract: We consider solutions to the nonlinear eigenvalue problem

$\displaystyle (*)\quad A(x,\vec u)\vec u + \lambda f(x,\vec u) = 0\:\quad {\tex... ...,\quad \vec u{\text{ = }}0,\quad {\text{on}}\partial \Omega ,\quad \vec{u} = 0,$

where (*) is a quasilinear strongly coupled second order elliptic system of partial differential equations and $ \Omega \subseteq \mathbf{R}^{n}$ is a smooth bounded domain. We obtain lower bounds for $ \lambda$ in the case where $ f(x,\vec u)$ has linear growth, and relations between $ \lambda ,\Omega $, and ess sup$ \vert\vec u\vert$ when $ f(x,\vec u)$ has sub- or superlinear growth. The estimates are based on integration by parts and application of certain Sobolev inequalities. We briefly discuss extensions to higher order systems.

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Article copyright: © Copyright 1984 American Mathematical Society