Existence of lattice solutions to semilinear elliptic systems with periodic potential

Under the assumption that the potential W is invariant under a general discrete reflection group $G'=TG$ acting on $mathbb{R}^n$, we establish existence of G'-equivariant solutions to $Delta u - W_u(u) = 0$, and find an estimate. By taking the size of the cell of the lattice in space domain to infinity, we obtain that these solutions converge to G-equivariant solutions connecting the minima of the potential W along certain directions at infinity. When particularized to the nonlinear harmonic oscillator $u''+alpha sin u=0$, $alpha>0$, the solutions correspond to those in the phase plane above and below the heteroclinic connections, while the G-equivariant solutions captured in the limit correspond to the heteroclinic connections themselves. Our main tool is the G'-positivity of the parabolic semigroup associated with the elliptic system which requires only the hypothesis of symmetry for W. The constructed solutions are positive in the sense that as maps from $mathbb{R}^n$ into itself leave the closure of the fundamental alcove (region) invariant.