Solutions of a system of diffusion equations.(English)Zbl 1134.35094

Summary: We study existence and multiplicity of homoclinic type solutions to the following system of diffusion equations on $$\mathbb R\times \Omega$$:
\begin{aligned} \partial_t u -\Delta_x u + b(t,x) \cdot \nabla_x u + V(x)u &= H_v (t,x,u,v), \\ -\partial_t v - \Delta_x v - b(t,x) \cdot \nabla_x v + V(x)v &= H_u (t,x,u,v),\end{aligned}
where $$\Omega = \mathbb R^N$$ or $$\Omega$$ is a smooth bounded domain of $$\mathbb R^N$$, $$z=(u,v): \mathbb R\times \Omega \rightarrow\mathbb R^m\times \mathbb R^m$$, and $$b \in {\mathcal C}^1(\mathbb R\times \overline{\Omega},\mathbb R^N)$$, $$V \in {\mathcal C}(\overline{\Omega},\mathbb R)$$, $$H \in {\mathcal C}^1(\mathbb R\times \overline{\Omega} \times \mathbb R^{2m},\mathbb R),$$ all three depending periodically on $$t$$ and $$x$$. We assume that $$H(t,x,0) \equiv 0$$ and $$H$$ is asymptotically quadratic or superquadratic as $$\mid z \mid \rightarrow \infty$$. The superquadratic condition is more general than the usual one. By establishing a proper variational setting based on some recent critical point theorems we obtain at least one nontrivial solution, and also infinitely many solutions provided $$H$$ is moreover symmetric in $$z$$.

MSC:

 35Q40 PDEs in connection with quantum mechanics 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis 35K45 Initial value problems for second-order parabolic systems 35A15 Variational methods applied to PDEs 37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010) 35B38 Critical points of functionals in context of PDEs (e.g., energy functionals)
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