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On semilinear elliptic boundary value problems in unbounded domains

Published online by Cambridge University Press:  14 February 2012

Deb Kumar Bose
Deb Kumar Bose
Affiliation:
Institut für Mathematik, Ruhr–Universität, Bochum
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Synopsis

We consider the possibility of solving semilinear elliptic boundary value problems in unbounded domains. We first treat the case when the non-linear terms are independent of terms involving gradients. Using a monotone iteration scheme, we show that the existence of a weak subsolution v and a weak supersolution w ≧ v, implies the existence of a weak solution u, and v ≦ u ≦ w. We also state conditions which guarantee the existence of a solution when only a subsolution is known to exist. Next, we suppose the non-linear terms can depend on gradient terms. Using a method developed in [4], based on perturbation theory of maximal monotone operators, we prove the existence of a H2(Ω) solution lying between a given H2(Ω) subsolution v and a given H2(Ω) supersolution w ≧ v.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 77 , Issue 3-4 , 1977 , pp. 177 - 192
Copyright
Copyright © Royal Society of Edinburgh 1977

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