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Non-existence of solutions for some nonlinear elliptic equations involving measures

Published online by Cambridge University Press:  11 July 2007

L. Orsina
Affiliation:
Dipartimento di Matematica, Università di Roma I, P.le A. Moro 2, 00185 Roma, Italy
A. Prignet
Affiliation:
Mathématiques, Université d'Orléans, 45067 Orléans Cedex 2, France

Abstract

In this paper, we study the non-existence of solutions for the following (model) problem in a bounded open subset Ω of RN: with Dirichlet boundary conditions, where p > 1, q > 1 and μ is a bounded Radon measure. We prove that if λ is a measure which is concentrated on a set of zero r capacity (p < rN), and if q > r (p − 1)/(rp), then there is no solution to the above problem, in the sense that if one approximates the measure λ with a sequence of regular functions fn, and if un is the sequence of solutions of the corresponding problems, then un converges to zero.

We also study the non-existence of solutions for the bilateral obstacle problem with datum a measure λ concentrated on a set of zero p capacity, with u in for every υ in K, finding again that the only solution obtained by approximation is u = 0.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2000

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