JOURNAL OF DIFFERENTIAL EQUATIONS 13, 247-256 (1973) Variational Methods for Nonlinear Eigenvalue Problems Associated with Thermal Ignition G. C. WAKE Mathematics Department, Victoria University, Wellington, New Zealand AND M. E. RAYNER St. Hilda’s College, Oxford, England Received November 11, 1971 1. INTRODUCTION The fallowing boundary value problem arises in the study of nonlinear heat generation (in the steady-state): w = Jw, a x = (Xl ) x2 )..., Xm) E D, (1) B(u) FEE cx(x)tJ + p(x)(aup) = 0, XEDD, (2) where L is the uniformly elliptic, self-adjoint, second-order operator and D is the interior of a bounded region of R” with a smooth boundary 80. The coefficients ai = aji(x) are continuously differentiable, a,(x) > 0 is continuous, and for all unit vectors p = (p, , p, )..., pm), it, PM4 $3 > 0, x E D. In Eq. (2) alaY is the conormal derivative: 247 Copyright 0 1973 by Academic Press, Inc. All rights of reproduction in any form reserved.