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

By Marius Ghergu, Vicentiu D. Rădulescu. Proceedings of the Royal Society of Edinburgh. Section A Mathematics, 2005

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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.

Subfields: Applied Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/6380668076a08df31b7f6ecc2651470c0a5b72a6

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Citations in the corpus

Lane-Emden-Fowler equations with convection and singular potential (2007, background): “Also, we proved in [17] that if p ≡ 1, then the existence of a classical solution to (P )− does not depend on the asymptotic behavior of g near the origin, whereas the exponent a of the convection term |∇u|a plays a crucial role.”
Singular phenomena in nonlinear elliptic problems. From blow-up boundary solutions to equations with singular nonlinearities (2006, result): “Our main results (see Ghergu and Rădulescu [47]) are the following.”
Nonlinear nonhomogeneous singular problems (2019, result): “Previously, such bifurcation type results have been proved by Sun, Wu & Long [41], Haitao [18], Ghergu & Rădulescu [14, 15, 16] (semilinear problems driven by the Laplacian), and by Giacomoni, Schindler & Takač [17], Papageorgiou, Rădulescu & Repovš [35], Papageorgiou & Smyrlis [36], Papageorgiou, V”
Ground state solutions for the singular Lane-Emden-Fowler equation with sublinear convection term (2006, background): “The influence of the convection term has been emphasized in Ghergu and Rădulescu [15, 16] and Zhang [26].”