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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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