We consider the problemwhere Ω is a smooth and bounded domain in R 2 and V is a positive smooth function in Ω. Let Γ be a closed, non-degenerate geodesic with respect to the metric ds 2 = V (y)(dy 2 1 + dy 2 2 ) in Ω.We prove that there exist two interior transition layer solutions u(1) ε , u (2) ε when ε is sufficiently small. One of the layer solutions u (1) ε approaches −1 in Ω 1 and +1 in Ω 2 = Ω\Ω 1 as ε tends to 0, while the other solution u(2) ε exhibits a transition layer in the opposite direction.
International audienceWe consider the problem e(2s) (-partial derivative(xx))(s)u(x) - V (x) over bar (u)over bar(x)over bar (1-u(2))((x)over bar) = 0 where (-partial derivative xx)(s) denotes the usual fractional Laplace operator, epsilon > 0 is a small parameter and the smooth bounded function V satisfies infi x is an element of R V (x) > 0. For a E 1), we prove the existence of separate multi layered solutions for any small a, where the layers are located near any non -degenerate local maximal points and non-degenerate local minimal points of function V. We also prove the existence of clustering-layered solutions, and these clustering layers appear within a very small neighborhood of a local maximum point of V
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