Periodic solutions for nonresonant parabolic equations on $${\mathbb {R}}^N$$ with Kato–Rellich type potentials
Tóm tắt
A criterion for the existence of T-periodic solutions of nonautonomous parabolic equation $$u_t = \Delta u + V(x)u + f(t,x,u)$$ , $$x\in {\mathbb {R}}^N$$ , $$t>0$$ , 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.
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