The N-Laplacian elliptic equation: Variational versus entropy solutions

Abstract

The N-Laplacian equation -∇ · (\∇u\N - 2∇u) = f is a kind of limit case in the existence theory of nonlinear elliptic equations when the second member is assumed to be merely integrable (L1 theory). Here we consider for definiteness the solutions of the homogeneous Dirichlet problem in a bounded domain Ω ∈ RN and compare the standard concept of a variational (or energy) solution with the recently introduced concept of an entropy solution, which seems natural in the L1 theory. For our equation both concepts have slightly different domains of application. We also discuss the conditions on f under which the solutions are bounded, a slightly smaller class. © 1996 Academic Press, Inc.