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An eigenvalue problem for generalized Laplacian in Orlicz—Sobolev spaces

Published online by Cambridge University Press:  14 November 2011

Vesa Mustonen  and
Matti Tienari
Vesa Mustonen
Affiliation:
Department of Mathematical Sciences, University of Oulu, FIN 90570, Oulu, Finland (vesa.mustonen@oulu.fi
Matti Tienari
Affiliation:
Department of Mathematical Sciences, University of Oulu, FIN 90570, Oulu, Finlandmatti.tienari@oulu.fi
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Let m: [ 0, ∞) → [ 0, ∞) be an increasing continuous function with m(t) = 0 if and only if t = 0, m(t) → ∞ as t → ∞ and Ω C ℝN a bounded domain. In this note we show that for every r > 0 there exists a function ur solving the minimization problem

where Moreover, the function ur is a weak solution to the corresponding Euler–Lagrange equation

for some λ > 0. We emphasize that no Δ2-condition is needed for M or M; so the associated functionals are not continuously differentiable, in general.

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Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 129 , Issue 1 , 1999 , pp. 153 - 163
Copyright
Copyright © Royal Society of Edinburgh 1999

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