Range of semilinear operators for systems at resonance

Abstract

For a vector function u : R → RN we consider the system u 00(t) + ∇G(u(t)) = p(t) u(t) = u(t + T), where G : RN → R is a C1 function. We are interested in finding all possible T-periodic forcing terms p(t) for which there is at least one solution. In other words, we examine the range of the semilinear operator S : H2 per → L2 ([0, T], RN ) given by Su = u 00 + ∇G(u), where H2 per = {u ∈ H2 ([0, T], R N ); u(0) − u(T) = u 0 (0) − u 0 (T) = 0}. Writing p(t) = p + pe(t), where p := 1 T R T 0 p(t) dt, we present several results concerning the topological structure of the set I(pe) = {p ∈ R N ; p + pe ∈ Im(S)}.