Existence Results for a Perturbed Problem Involving Fractional Laplacians
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 s...
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Wiley
2014-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2014/548301 |
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| author | Yan Hu |
| author_facet | Yan Hu |
| author_sort | Yan Hu |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-afacbef6d413427484a347ca3b761278 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-afacbef6d413427484a347ca3b7612782025-02-03T01:24:08ZengWileyAbstract and Applied Analysis1085-33751687-04092014-01-01201410.1155/2014/548301548301Existence Results for a Perturbed Problem Involving Fractional LaplaciansYan Hu0College of Mathematics and Econometrics, Hunan University, Changsha 410082, ChinaWe 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.http://dx.doi.org/10.1155/2014/548301 |
| spellingShingle | Yan Hu Existence Results for a Perturbed Problem Involving Fractional Laplacians Abstract and Applied Analysis |
| title | Existence Results for a Perturbed Problem Involving Fractional Laplacians |
| title_full | Existence Results for a Perturbed Problem Involving Fractional Laplacians |
| title_fullStr | Existence Results for a Perturbed Problem Involving Fractional Laplacians |
| title_full_unstemmed | Existence Results for a Perturbed Problem Involving Fractional Laplacians |
| title_short | Existence Results for a Perturbed Problem Involving Fractional Laplacians |
| title_sort | existence results for a perturbed problem involving fractional laplacians |
| url | http://dx.doi.org/10.1155/2014/548301 |
| work_keys_str_mv | AT yanhu existenceresultsforaperturbedprobleminvolvingfractionallaplacians |