Riv. Mat. Univ. Parma, Vol. 5, No. 1, 2014

Matteo Bonforte [1] and Agnese Di Castro [2][3]

Quantitative local estimates for nonlinear elliptic equations involving p-Laplacian type operators

Pages: 213-271
Received: 23 July 2013   
Accepted in revised form: 12 March 2014
Mathematics Subject Classification (2010): 35B45, 35B65, 35J60, 35J61.

Keywords: Nonlinear elliptic equations of p-Laplacian type, local bounds, Harnack inequalities.
Authors addresses:
[1] : Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco, Madrid, 28049, Spain
[2] : Dipartimento di Matematica e Informatica, UniversitÓ degli Studi di Parma, Campus - Parco Area delle Scienze, 53/A, Parma, 43124, Italy
[3] : Dipartimento di Matematica, UniversitÓ di Pisa, Largo Bruno Pontecorvo 5, Pisa 56127, Italy

Abstract: The purpose of this paper is to prove quantitative local upper and lower bounds for weak solutions of elliptic equations of the form \(-\Delta_p u= \lambda u^s\), with \(p>1\), \(s\geq0\) and \(\lambda\geq 0\), defined on bounded domains of \(\mathbb{R}^d\), \(d\ge 1\), without reference to the boundary behaviour. We give an explicit expression for all the involved constants. As a consequence, we obtain local Harnack inequalities with explicit constants. Finally, we discuss the issue of local absolute bounds, which are new to our knowledge. Such bounds will be true only in a restricted range of \(s\) or for a special class of weak solutions, namely for local stable solutions. In the study of local absolute bounds for stable solutions there appears the so-called Joseph-Lundgren exponent as a limit of applicability of such bounds.

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