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Multi-parameter bifurcation and asymptotics for the singular Lane–Emden–Fowler equation with a convection term

Published online by Cambridge University Press:  12 July 2007

Marius Ghergu
Affiliation:
Department of Mathematics, University of Craiova, 200585 Craiova, Romania ([email protected])
Vicenţiu Rădulescu
Affiliation:
Department of Mathematics, University of Craiova, 200585 Craiova, Romania ([email protected])
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Abstract

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We establish some bifurcation results for the boundary-value problem −Δu = g (u) + λ|∇u|p + μf (x, u) in Ω, u > 0 in Ω, u = 0 on δΩ, where Ω is a smooth bounded domain in RN, λ, μ ≥ 0, 0 < p ≤ 2, f is non-decreasing with respect to the second variable and g is unbounded around the origin. The asymptotic behaviour of the solution around the bifurcation point is also established, provided g(u) behaves like u−α around the origin, for some 0 < α < 1. Our approach relies on finding explicit sub- and supersolutions combined with various techniques related to the maximum principle for elliptic equations. The analysis we develop in this paper shows the key role played by the convection term |∇u|p.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2005