Abstract
We consider a class of quasilinear elliptic equations of Leray-Lions type that allow us to control the ess inf and ess sup of solutions. Precisely, for arbitrary given four constants m 0 < m 1 ≤ m 1 < m 0, we find some sufficient conditions such that for each solution u ε W 1, p (Ω) satisfying m 0 ≤ u( x) ≤ M 0 on ∂Ω, we have m 0 ≤ ess inf u < m 1 and m 1 < ess sup u ≤ M 0. The main consequence of this result is the lower bound of oscillation of solutions. It enables us to generate singular solutions and to obtain a lower bound on the constants in Schauder and Agmon, Douglis, Nirenberg estimates.
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