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

Abstract

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 ∂Ω, 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. © 2005 The Royal Society of Edinburgh.