Instability of nonnegative solutions for a class of semipositone problems

We consider the boundary value problem -Au(x) = Af (u(x)), x E Q Bu(x) = O, x E al where Q is a bounded region in RN with smooth boundary, Bu = ah(x)u + (1 a)au/an where a E [0, 1] h: a -+ R with h = 1 when a = 1, A > 0, f is a smooth function with f(O) 0 for u > 0 and f"(u) > 0 for u > 0. We prove that every nonnegative solution is unstable. INTRODUCTION In this paper we consider the stability of the semilinear elliptic boundary value problem (1.1) -Au(x) = if (u(x)), x E Q, (1.2) Bu(x) = 0, x E OQ, where Q is a bounded region in R Nwith smooth boundary, Bu(x) = ah(x)u(x) + (1 a)Ou/On where a E [0, 1] is a constant, h OQ a R+ is a smooth function with h _ 1 when a = 1, i.e., the boundary condition may be of Dirichlet, Neumann or mixed type, A > 0 is a constant and f is a smooth function satisfying: (1.3) f(O) 0 for u>0, (1.5) f'(u) > 0 for u > 0. Recall that a solution of (1.1)-(1.2) is stable in the maximum norm if given any e > 0 there exists a J5 > 0 such that if lluo(x) U(x)llO < c5 then Received by the editors October 26, 1989 and, in revised form, February 28, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 35B35, 35J65. Part of this work was carried out while R. Shivaji was visiting Heriot-Watt University as part of an SERC funded research program and also supported in part by NSF Grant DMS-8905936. ? 1991 American Mathematical Society 0002-9939/91 $1.00+ $.25 per page