Layer solutions for a class of semilinear elliptic equations involving fractional Laplacians

This paper is concerned with the nonlinear equation involving the fractional Laplacian: (-Delta)(s)v(x) = b(x)f(v(x)), x is an element of R, where s is an element of (0, 1), b : R -> R is a periodic, positive, even function and -f is the derivative of a double-well potential G. That is, G is an element of C-2,C-gamma (0 < gamma < 1), G(1) = G(-1) < G(tau) for all tau is an element of (-1, 1), G'(-1) = G'(1) = 0. We show the existence of layer solutions of the equation for s >= 1/2 and for some odd nonlinearities by variational methods, which is a bounded solution having the limits +/- 1 at +/-infinity. Asymptotic estimates for layer solutions as vertical bar x vertical bar -> +infinity and the asymptotic behavior of them as s up arrow 1 are also obtained.