Number of Solutions with a Norm Bounded by a Given Constant of a Semilinear Elliptic PDE with a Generic Right-Hand Side

Abstract

Abstract.We consider a semilinear boundary value problem -Au+f(u, x) = 0 in ß C S.N and u = 0 on dQ.. We assume that / is a C°°-smooth function and Í2 is a bounded domain with a smooth boundary. For any Csmooth perturbation h(x) of the right-hand side of the equation we consider the function Nh(S) defined as the number of C2+a-smooth solutions u such that ||«||C0(Q) < 5 of the perturbed problem.How "small" JVft(S) can be made by a perturbation h(x) suchthat P||co/q) < e ? We present here an explicit upper bound in terms of e , S and max _||D¿/(ii,x)|| (¿€{0,1,2}). \u\<s,xenIf S is fixed then h can be chosen by such a way that the upper bound persists under small in C°-topology perturbations of h . We present an explicit lower bound for the radius of the ball of such admissible perturbations.