Qualitative properties of positive solutions of semilinear elliptic equations in symmetric domains via the maximum principle

In this paper we study the positive solutions of the equation -Delta u + la mbda u = f(u) in a bounded symmetric domain Omega in R-N, with the boundary condition u = 0 on partial derivative Omega. Using the maximum principle w e prove the symmetry of the solutions v of the linearized problem. From thi s we deduce several properties of v and u; in particular we show that if N = 2 there cannot exist two solutions which have the same maximum if f is al so convex and that there exists only one solution if f(u) = u(p) and lambda = 0. In the final section we consider the problem -Delta u = u(P) + mu u(q) in O mega with u = 0 on partial derivative Omega, and show that if 1 < p < N=2/N -2,q is an element of]0,1[ there are exactly two positive solutions for mu, sufficiently small and some particular domain Omega. (C) Elsevier, Paris.