Quantitative local estimates for nonlinear elliptic equations involving \(p\)-Laplacian type operators. (English) Zbl 1322.35042

Bonforte, Matteo; Di Castro, Agnese

Riv. Mat. Univ. Parma (N.S.) 5, No. 1, 213-271 (2014).

Authors’ 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 \(p>1\), \(s \geq 0\) and \(\lambda \geq 0\), defined on bounded domains of \(\mathbb{R}^d, d\geq 1\), 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.

Reviewer: Patrick Winkert (Berlin)

MSC:

35J92	Quasilinear elliptic equations with \(p\)-Laplacian
35B45	A priori estimates in context of PDEs
35B65	Smoothness and regularity of solutions to PDEs
35J60	Nonlinear elliptic equations
35J61	Semilinear elliptic equations
35D30	Weak solutions to PDEs

Keywords:

nonlinear elliptic equations of \(p\)-Laplacian type; local bounds; Harnack inequalities