Solutions of a system of diffusion equations

Abstract.

We study existence and multiplicity of homoclinic type solutions to the following system of diffusion equations on ${\mathbb{R}} \times \Omega$:

$$ \left\{ {\begin{array}{*{20}c} {\,\,{\partial}_t u - {\Delta}_x u + b(t,x) \cdot {\nabla}_x u + V(x)u = H_v (t,x,u,v),} \\ { - {\partial}_t v - {\Delta}_x v - b(t,x) \cdot {\nabla}_x v + V(x)v = H_u (t,x,u,v),}\\ \end{array} } \right. $$

where $\Omega = {\mathbb{R}}^N$ or Ω 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 moreever symmetric in z.