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Existence and non-existence of solutions to elliptic equations with a general convection term

Published online by Cambridge University Press:  20 March 2014

Salomón Alarcón
Affiliation:
Departamento de Matemática, Universidad Técnica Federico Santa María, Casilla V-110, Avenida España 1680, Valparaíso, Chile, (salomon.alarcon@usm.cl)
Jorge García-Melián
Affiliation:
Departamento de Análisis Matemático, Universidad de La Laguna, Campus de Anchieta, Avenida Astrofísico Francisco Sánchez s/n, 38271 La Laguna, Spain Instituto Universitario de Estudios Avanzados en Física Atómica, Molecular y Fotónica, Facultad de Física, Universidad de La Laguna, Campus de Anchieta, Avenida Astrofísico Francisco Saínchez s/n, 38203 La Laguna, Spain, (jjgarmel@ull.es)
Alexander Quaas
Affiliation:
Departamento de Matemaítica, Universidad Tíecnica Federico Santa María, Casilla V-110, Avenida Espanña 1680, Valparaíso, Chile, (, alexander.quaas@usm.cl)

Abstract

In this paper we consider the nonlinear elliptic problem −Δu + αu = g(∣∇u∣) + λh(x) in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain of ℝN, α ≥ 0, g is an arbitrary C1 increasing function and h ∈ C1() is non-negative. We completely analyse the existence and non-existence of (positive) classical solutions in terms of the parameter λ. We show that there exist solutions for every λ when α = 0 and the integral 1/g(s)ds = ∞, or when α > 0 and the integral s/g(s)ds = ∞. Conversely, when the respective integrals converge and h is non-trivial on ∂Ω, existence depends on the size of λ. Moreover, non-existence holds for large λ. Our proofs mainly rely on comparison arguments, and on the construction of suitable supersolutions in annuli. Our results include some cases where the function g is superquadratic and existence still holds without assuming any smallness condition on λ.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2014 

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