1 results
Entire solutions in ${\mathbb{R}}^{2}$
for a class of Allen-Cahn equations
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- Journal:
- ESAIM: Control, Optimisation and Calculus of Variations / Volume 11 / Issue 4 / October 2005
- Published online by Cambridge University Press:
- 15 September 2005, pp. 633-672
- Print publication:
- October 2005
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- Article
- Export citation
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We consider a class ofsemilinear elliptic equations of the form 15.7cm - $\varepsilon^{2}\Delta u(x,y)+a(x)W'(u(x,y))=0,\quad (x,y)\in{\mathbb{R}}^{2}$
where $\varepsilon>0$ 
, $a:{\mathbb{R}}\to{\mathbb{R}}$ 
is a periodic, positive function and $W:{\mathbb{R}}\to{\mathbb{R}}$ 
is modeled on the classical two well Ginzburg-Landaupotential $W(s)=(s^{2}-1)^{2}$ 
. We look for solutions to ([see full textsee full text])which verify theasymptotic conditions $u(x,y)\to\pm 1$ 
as $x\to\pm\infty$ 
uniformly with respect to $y\in{\mathbb{R}}$ 
.We show via variationalmethods that if ε is sufficiently small and a is not constant, then ([see full textsee full text])admits infinitely many of such solutions, distinct up to translations, which do not exhibit one dimensional symmetries.
