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 \(z\).


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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