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Solutions of p-Laplace Equations with Infinite Boundary Values: The case of Non-Autonomous and Non-Monotone Nonlinearities

Part of: Elliptic equations and systems Partial differential equations

Published online by Cambridge University Press:  22 December 2015

Michael A. Karls  and
Ahmed Mohammed
Michael A. Karls
Affiliation:
Department of Mathematical Sciences, Ball State University, 2000 W University Ave, Muncie, IN 47306, USA (mkarls@bsu.edu; amohammed@bsu.edu)
Ahmed Mohammed
Affiliation:
Department of Mathematical Sciences, Ball State University, 2000 W University Ave, Muncie, IN 47306, USA (mkarls@bsu.edu; amohammed@bsu.edu)
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Abstract

For a non-negative and non-trivial real-valued continuous function hΩ × [0, ∞) such that h(x, 0) = 0 for all x ∈ Ω, we study the boundary-value problem

where Ω ⊆ ℝN, N ⩾ 2, is a bounded smooth domain and Δp:= div(|Du|p–2DDu) is the p-Laplacian. This work investigates growth conditions on h(x, t) that would lead to the existence or non-existence of distributional solutions to (BVP). In a major departure from past works on similar problems, in this paper we do not impose any special structure on the inhomogeneous term h(x, t), nor do we require any monotonicity condition on h in the second variable. Furthermore, h(x, t) is allowed to vanish in either of the variables.

Keywords

boundary blow-up solutionminimality principlep-Laplacian

MSC classification

Secondary: 35B44: Blow-up 35J92: Quasilinear elliptic equations with $p$-Laplacian 35J25: Boundary value problems for second-order elliptic equations
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 4 , November 2016 , pp. 959 - 987
Copyright
Copyright © Edinburgh Mathematical Society 2016 

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