Infinitely many solutions for diffusion equations without symmetry (Q622400)

The paper is concerned with the existence of homoclinic solutions \(z=(u,v):\mathbb{R}\times\mathbb{R}^N\to\mathbb{R}^m\times\mathbb{R}^m\) of the system \[ \left\{ \begin{aligned} \partial_t u-\Delta_x u+b(t,x)\nabla_xu+V(x)u= H_v(t,x,u,v)\\ -\partial_t v-\Delta_x v+b(t,x)\nabla_x v+V(x)v= H_u(t,x,u,v). \end{aligned} \right.\tag{HS} \] (HS) can be considered as an unbounded Hamiltonian system in \(L^2(\mathbb{R}\times\mathbb{R}^N, \mathbb{R}^{2m})\). It is assumed that \(b,V,H\) are periodic with respect to \(t\) and \(x_j\), \(j=1,\ldots, N\). If these functions satisfy various additional growth and sign conditions, the existence of infinitely many geometrically different homoclinic solutions is proved. The proof is based on variational methods for strongly indefinite functionals and uses an idea of \textit{N. Ackermann} [J. Funct. Anal. 234, No. 2, 277--320 (2006; Zbl 1126.35057)] to obtain multibump solutions.