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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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
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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| Summary: | 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. |
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| ISSN: | 1085-3375 1687-0409 |