THE N-LAPLACIAN ELLIPTIC EQUATION - VARIATIONAL VERSUS ENTROPY SOLUTIONS

The N-Laplacian equation -del .(\del u\(N-2)del u)=f is a kind of limi t case in the existence theory of nonlinear elliptic equations when th e second member is assumed to be merely integrable (L(1) theory). Here we consider for definiteness the solutions of the homogeneous Dirichl et problem in a bounded domain Omega is an element of R(N) and compare the standard concept of a variational (or energy) solution with the r ecently introduced concept of an entropy solution, which seems natural in the L(1) theory. For our equation both concepts have slightly diff erent domains of application. We also discuss the conditions on f unde r which the solutions are bounded, a slightly smaller class. (C) 1996 Academic Press, Inc.