Abstract

For Ω, an open bounded subset of RN with smooth boundary and 1<p<∞, we establish W1,p(Ω) a priori bounds and prove the compactness of solution sets to differential inequalities of the form which are bounded in L∞(Ω). The main point in this work is that the nonlinear term F may depend on ∇u and may grow as fast as a power of order p in this variable. Such growth conditions have been used extensively in the study of boundary value problems for nonlinear ordinary differential equations and are known as Bernstein-Nagumo growth conditions. In addition, we use these results to establish a sub-supersolution theorem.