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

The purpose of this paper is to prove quantitative local upper and lower bounds for weak solutions of elliptic equations of the form -Delta pu = lambda u(s), with p > 1, s >= 0 and lambda >= 0, defined on bounded domains of R-d, d >= 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.