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A critical Kirchhoff type problem involving a nonlocal operator
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TLDR
In this paper, the existence of non-negative solutions for a Kirchhoff type problem driven by a non-local integrodifferential operator is shown, where L K is an integro-differential operator with kernel K, Ω is a bounded subset of R n, M and f are continuous functions, and 2 ∗ is a fractional Sobolev exponent.Abstract:
In this paper we show the existence of non-negative solutions for a Kirchhoff type problem driven by a nonlocal integrodifferential operator, that is − M ( ‖ u ‖ Z 2 ) L K u = λ f ( x , u ) + | u | 2 ∗ − 2 u in Ω , u = 0 in R n ∖ Ω where L K is an integrodifferential operator with kernel K , Ω is a bounded subset of R n , M and f are continuous functions, ‖ ⋅ ‖ Z is a functional norm and 2 ∗ is a fractional Sobolev exponent.read more
Citations
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Book
Variational Methods for Nonlocal Fractional Problems
TL;DR: A thorough introduction to the variational analysis of nonlinear problems described by nonlocal operators can be found in this paper, where the authors give a systematic treatment of the basic mathematical theory and constructive methods for these classes of equations, plus their application to various processes arising in the applied sciences.
Journal ArticleDOI
Multiple solutions for nonhomogeneous Schrödinger–Kirchhoff type equations involving the fractional p -Laplacian in $${\mathbb {R}}^N$$ R N
TL;DR: In this article, the existence of multiple solutions for the nonhomogeneous fractional p-Laplacian equations of Schrodinger-Kirchhoff type was investigated, and multiplicity results were obtained by using the Ekeland variational principle and the Mountain Pass theorem.
Journal ArticleDOI
Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity
TL;DR: In this article, the existence and the asymptotic behavior of non-negative solutions for a class of stationary Kirchhoff problems driven by a fractional integro-differential operator LK and involving a critical nonlinearity were analyzed.
Journal ArticleDOI
Superlinear Schrödinger–Kirchhoff type problems involving the fractional p–Laplacian and critical exponent
TL;DR: In this paper, the existence and multiplicity of solutions for the Schrődinger-Kirchhoff type problems involving the fractional p-Laplacian and critical exponent were studied.
Journal ArticleDOI
Existence of solutions for Kirchhoff type problem involving the non-local fractional p-Laplacian
TL;DR: In this paper, the existence of weak solutions for a Kirchhoff type problem driven by a non-local integro-differential operator of elliptic type with homogeneous Dirichlet boundary conditions was investigated.
References
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TL;DR: In this article, the theory of conjugate convex functions is introduced, and the Hahn-Banach Theorem and the closed graph theorem are discussed, as well as the variations of boundary value problems in one dimension.
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TL;DR: In this article, the authors deal with the fractional Sobolev spaces W s;p and analyze the relations among some of their possible denitions and their role in the trace theory.
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Mountain Pass solutions for non-local elliptic operators
TL;DR: In this article, the existence of solutions for equations driven by a non-local integrodifferential operator with homogeneous Dirichlet boundary conditions was studied and a nonlinear solution for them using the Mountain Pass Theorem was found.
Journal ArticleDOI
Variational methods for non-local operators of elliptic type
TL;DR: In this paper, the existence of non-trivial solutions for the problem driven by a non-local integrodifferential operator with homogeneous Dirichlet boundary conditions was studied.