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

Abstract

A criterion for the existence of T-periodic solutions of nonautonomous parabolic equation $$u_t = \Delta u + V(x)u + f(t,x,u)$$ u t = Δ u + V ( x ) u + f ( t , x , u ) , $$x\in {\mathbb {R}}^N$$ x ∈ R N , $$t>0$$ 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. $${\mathrm {Ker}} (\Delta + V)=\{0\}$$ Ker ( Δ + V ) = { 0 } , the equation admits a T-periodic solution. Moreover, in case there is a trivial branch of solutions, i.e. $$f(t,x,0)=0$$ f ( t , x , 0 ) = 0 , there exists a nontrivial solution provided the total multiplicities of positive eigenvalues of $$\Delta +V$$ Δ + V and $$\Delta + V + f_0$$ Δ + V + f 0 , where $$f_0$$ f 0 is the partial derivative $$f_u(\cdot ,\cdot ,0)$$ f u ( · , · , 0 ) of f, are different mod 2.