MULTIPLICITY OF SOLUTIONS OF A QUASI-LINEAR ELLIPTIC EQUATION IN SPHERICAL DOMAINS

The quasilinear equation, which arises in combustion theory in the inv estigation of the steady-state energy balance DELTAu + lambdae(u) = 0, x is-an-element-of OMEGA subset-or-equal-to R(n) partial derivative u /partial derivative n + Bi.u = 0, x is-an-element-of partial derivativ e OMEGA has an intriguing solution set in spherical domains where it b ecomes d2u/dr2 + n-1/p du/dr + lambdae(u) = 0, 0 < r < 1 u'(1) + Bi.u( 1) = 0, u'(0) = 0. This can be shown to have an infinite number of fin ite positive solutions when lambda = lambda(infinity) = 2(n - 2)e-2/Bi , when 2 < n < 10. Phase plane techniques are used. All but the minima l solutions are unstable as solutions of the time-dependent version of the above equations. The extension of these methods to spherically an nular domains 0 < alpha < r < 1, with an inner boundary condition u'(a lpha) = A(less-than-or-equal-to 0), shows strikingly different behavio ur. First, the infinite multiplicity disappears and is approached asym ptotically as alpha --> 0+. Second, the uniqueness of the solution for small lambda also disappears. This last fact has implications for the basis of stability of the minimal steady-state. The case A not-equal 0 is still under investigation. Collaborators in this work are P. Kell y (Massey University) and S. K. Scott (University of Leads).