Existence Results for a Perturbed Problem Involving Fractional Laplacians

Abstract EN

We extend the results of Cabre and Sire (2011) to show the existence of layer solutions of fractional Laplacians with perturbed nonlinearity (-Δ)su=b(x)f(u) in ℝ with s∈(0,1).

Here b is a positive periodic perturbation for f, and -f is the derivative of a balanced well potential G.

That is, G∈C2,γ satisfies G(1)=G(-1)

First, for odd nonlinearity f and for every s∈(0,1), we prove that there exists a layer solution via the monotone iteration method.

Besides, existence results are obtained by variational methods for s∈(1/2,1) and for more general nonlinearities.

While the case s≤1/2 remains open.