Existence results for a perturbed problem involving fractional Laplacians
Summary: We extend the results of \textit{X. Cabré} and \textit{Y. Sire} [Trans. Am. Math. Soc. 367, No. 2, 911--941 (2015; Zbl 1317.35280), \url{arXiv:1111.0796}] to show the existence of layer solutions of fractional Laplacians with perturbed nonlinearity \((- \Delta)^s u = b(x) f(u)\) in \(\mathbb{R}\) with \(s \in(0,1)\). Here \(b\) is a positive periodic perturbation for \(f\), and \(- f\) is the derivative of a balanced well potential \(G\). That is, \(G \in C^{2, \gamma}\) satisfies \(G(1) = G(- 1) < G(\tau) \forall \tau \in(- 1,1), G'(1) = G'(- 1) = 0 \). First, for odd nonlinearity \(f\) and for every \(s \in(0,1)\), we prove that there exists a layer solution via the monotone iteration method. Besides, existence results are obtained by variational methods for \(s \in(1 / 2, 1)\) and for more general nonlinearities. While the case \(s \leq 1 / 2\) remains open.
scientific article; zbMATH DE number 7022604