**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, Università degli Studi di Parma, Campus - Parco Area delle Scienze, 53/A, Parma, 43124, Italy

[3] : Dipartimento di Matematica, Università 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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