Electron. J. Diff. Equ., Vol. 2012 (2012), No. 15, pp. 1-15.

Existence of lattice solutions to semilinear elliptic systems with periodic potential

Nicholas D. Alikakos, Panayotis Smyrnelis

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.

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Nicholas D. Alikakos
Department of Mathematics, University of Athens
Panepistemiopolis, 15784 Athens, Greece
email: nalikako@math.uoa.gr
Panayotis Smyrnelis
Department of Mathematics
Aristotle University of Thessaloniki
54124 Thessaloniki, Greece
email: ysmyrnelis@yahoo.fr

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