WEAK EIGENFUNCTIONS FOR THE LINEARIZATION OF EXTREMAL ELLIPTIC PROBLEMS

We consider the semilinear elliptic problem [GRAPHICS] where lambda is a nonnegative parameter and g is a positive, nondecreasing, convex no nlinearity. There exists a value lambda of the parameter which is ext remal in terms of existence of solution. We study the linearization of the semilinear problem at the extremal weak solution corresponding to the parameter lambda=lambda. In some cases, this linearized problem has discrete and positive H-0(1)-spectrum. However, we prove that ther e always exists a positive weak eigenfunction in L-1(Omega) with eigen value zero for this linearized problem. The zero L-1-eigenvalue is coh erent with thee nonexistence of solutions of the semilinear problem fo r lambda > lambda. Finally, we find all weak eigenfunctions and eigen values for the linearization of the extremal problem when Omega is the unit ball and g(u) = e(u) or g(u)= (1 + u)(p). (C) 1998 Academic Pres s.