Control of essential infimum and supremum of solutions of quasilinear elliptic equations

We consider a class of quasilinear elliptic equations of Leray-Lions type t hat allow us to control the ess inf and ess sup of solutions. Precisely, fo r arbitrary given four constants m(0) < m(1) less than or equal to M-1 < M- 0, we find some sufficient conditions such that for each solution u is an e lement of W-1,W-p(Omega) satisfying m(0) less than or equal to u(x) less th an or equal to M-0 on delta Omega, we have m(0) less than or equal to ess(O mega)inf u < m(1) and M-1 < ess(Omega)sup u less than or equal to M-0. The main consequence of this result is the lower bound of oscillation of soluti ons. It enables us to generate singular solutions and to obtain a lower bou nd on the constants in Schauder and Agmon, Douglis, Nirenberg estimates. (C ) 1999 Academie des Sciences / Editions scientifiques et medicales Elsevier SAS.