Existence Results for a Perturbed Problem Involving Fractional Laplacians - Yan Hu (Abstract and Applied Analysis)
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)<G(τ)  ∀τ∈(-1,1),  G'(1)=G'(-1)=0. 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.