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    188宝金博页面版: Existence Results for Some Partial Integro-Differential Equations

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    内容提示： Mediterr. J. Math. 13 (2016), 4063–4079DOI 10.1007/s00009-016-0732-61660-5446/16/064063-17published onlineMay 19, 2016c ? Springer International Publishing 2016Existence Results for Some PartialIntegro-Dif f erential EquationsSenoussi Guesmia, Rokia Kechkar and Mohamed Said MoulayAbstract. In this note, we deal with semilinear integro-dif f erential equa-tions subject to homogeneous Dirichlet boundary conditions given onthe boundaries of the sections. Even if the dif f erentiation will be takenonly in som...

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    Mediterr. J. Math. 13 (2016), 4063–4079DOI 10.1007/s00009-016-0732-61660-5446/16/064063-17published onlineMay 19, 2016c ? Springer International Publishing 2016Existence Results for Some PartialIntegro-Dif f erential EquationsSenoussi Guesmia, Rokia Kechkar and Mohamed Said MoulayAbstract. In this note, we deal with semilinear integro-dif f erential equa-tions subject to homogeneous Dirichlet boundary conditions given onthe boundaries of the sections. Even if the dif f erentiation will be takenonly in some directions, it is not possible to see the main problem para-meterized by the other coordinates because of the non-local terms whichalso obliged the problem to be degenerate. We establish the existenceof solutions by employing the singular perturbations method as a nat-ural tool. The perturbed problems are classical, non-local, semilinearelliptic problems and the limits of the subsequences of their solutions,in weighted Sobolev type spaces, are solutions of the main problem.Some improvement, concerning the existence of the solutions and theconvergence results depending on the weights, will be established. Thepaper also gives an idea about the study of the anisotropic singularperturbations in the framework of weighted spaces.Mathematics Subject Classif i cation. Primary 35B25, 35B40, 35J70,35R09; Secondary 35J61, 45K05.Keywords. Semilinear and degenerate elliptic equations, anisotropic,integro-dif f erential problems, singular perturbations, asymptotic behav-ior.1. Introduction and Position of the ProblemLet Ω = ω 1 × ω 2 ⊂ R p × R n be a bounded open subset and n,p be positiveintegers. We set x = (X 1 ,X 2 ) such that X 1 ∈ ω 1 and X 2 ∈ ω 2 , that is to say,X 1 = (x 1 ,...,x p ), X 2 = (x ? 1 ,...,x ? n ).With these notations we set∇u =? ∂x 1 u,...,∂ x p u,∂ x ?1 u,...,∂ x ? n u? T=?∇ X 1 u∇ X 2 u?,S. Guesmia has been supported by the university of Qassim under the contract SR-D-2219/1434.

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