Instability of nonnegative solutions for a class of semilinear elliptic boundary value problems

We consider the boundary value problem -Delta u(x)=lambda f(u(x)), x is an element of Omega, But(x)=0, x is an element of partial derivative Omega, where Omega is a bounded region in R-n with smooth boundary Bu(x)=alpha h(x)u + (1 - alpha)partial derivative u/partial derivative n where alpha is an element of [0, 1], h : partial derivative Omega --> R+ with h = 1 when alpha = 1,lambda > 0, f is a smooth function such that f "(u) > 0 for u > 0, flu) < 0 for u is an element of (0,beta) and f(u) > 0 for u > beta for some beta > 0. We provide a simple proof to establish that every non-trivial nonnegative solution is unstable. (C) 1998 Elsevier Science B.V. All rights reserved.