Periodic solutions for nonresonant parabolic equations on {mathbb {R}}^N with Kato\u2013Rellich type potentials

Abstract

A criterion for the existence of T-periodic solutions of nonautonomous parabolic equation u_t = Delta u + V(x)u + f(t,x,u), xin {mathbb {R}}^N, t>0, where V is Kato–Rellich type potential and f diminishes at infinity, will be provided. It is proved that, under the nonresonance assumption, i.e. {mathrm {Ker}} (Delta + V)={0}, the equation admits a T-periodic solution. Moreover, in case there is a trivial branch of solutions, i.e. f(t,x,0)=0, there exists a nontrivial solution provided the total multiplicities of positive eigenvalues of Delta +V and Delta + V + f_0, where f_0 is the partial derivative f_u(cdot ,cdot ,0) of f, are different mod 2.