Existence Results for a Perturbed Problem Involving Fractional Laplacians

Abstract

We extend the results of Cabre and Sire (2011) to show the existence of layer solutions of fractional Laplacians with perturbed nonlinearity $(-\mathrm{\Delta }{)}^{s}u=b(x)f(u)$ in $ℝ$ with $s\in \mathrm{(0,1)}$. Here $b$ is a positive periodic perturbation for $f$, and $-f$ is the derivative of a balanced well potential $G$. That is, $G\in C^{\mathrm{2},\gamma }$ satisfies First, for odd nonlinearity $f$ and for every $s\in (\mathrm{0,1})$, we prove that there exists a layer solution via the monotone iteration method. Besides, existence results are obtained by variational methods for $s\in (\mathrm{1}/\mathrm{2},\mathrm{}\mathrm{1})$ and for more general nonlinearities. While the case $s\le \mathrm{1}/\mathrm{2}$ remains open.