Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 15, pp. 1-15.

Title: Existence of lattice solutions to semilinear elliptic systems
       with periodic potential

Authors: Nicholas D. Alikakos (Univ. of Athens, Greece)
         Panayotis Smyrnelis (Aristotle Univ., Thessaloniki, Greece)

Abstract:
 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.
 
Submitted October 24, 2011. Published January 24, 2012.
Math Subject Classifications: 35J20, 35J50.
Key Words: Lattice solution; invariant; periodic Potential; elliptic system; 
           reflection; discrete group; variational calculus.