JOURNAL OF DIFFERENTIAL EQUATIONS J6, 270-289 (1985) Bounds for the Spectrum and the Norm of Solutions of Nonlinear Elliptic Equations and Systems C. COSNER Department of Mathematics, University of Miami, Coral Gables, Florida 33124 AND M. H. PROTTER* Department of Mathematics, University of California, Berkeley, California 94720 Received August 12, 1983 1. INTRODUCTION Let 52 c R” be a bounded domain with smooth boundary and let I be a parameter. We consider the eigenvalue problem ECU] + J&x, u) = 0 in 52 u=o on i3C2 where E[u]= f -c i,j= 1axi ( %j(x, u,Vu) g t > (1.1) (1.2) is an elliptic operator in 8. For solutions of (1.1) which are positive in 52,it is of interest to estimate the size of the norm of the eigenfunction in terms of the corresponding eigenvalue. Payne and Stakgold [7] established such inequalities when E is the Laplacian and the nonlinear function f depends on u alone. In Section 2 we establish upper and lower bounds for eigen- values corresponding to nonnegative solutions of (1.1) as well as inequalities which yield, in the nonlinear case, estimates for the maximum value of an eigenfunction in terms of its eigenvalue. * Partially supported by the Office of Naval Research under Contract NOOO14-76-0316. 270 OO22-0396/85 $3.00 Copyright 0 1985 by Academic Press, Inc All rights of reproduction in any form reserved.