Existence results for a perturbed problem involving fractional Laplacians. (English) Zbl 1475.35387

Summary: We extend the results of X. Cabré and Y. Sire [Trans. Am. Math. Soc. 367, No. 2, 911–941 (2015; Zbl 1317.35280), 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.


35R11 Fractional partial differential equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian


Zbl 1317.35280
Full Text: DOI


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