Variational methods for non-local operators of elliptic type

By Raffaella Servadei, Enrico Valdinoci. 2012

738 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/e39ca3ffd76215cb64c994e2797424af153a2e6c

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Existence of solutions for Kirchhoff type problem involving the non-local fractional p-Laplacian (2015, background): “We refer also to [13,14,29,30,32,40,43,44] for further details on the functional framework and its applications to the existence of solutions for problem (1.”
A Brezis-Nirenberg type result for mixed local and nonlocal operators (2022, neutral): “…as in Remark 3.1-(4), we have for some C > 0, therefore (up to renaming C line after line) Moreover, recalling that by arguing as in the proof of Theorem 1.1 we have From this, since U = U 1 , 0 is an optimal function in the classical Sobolev inequality (and since s < 1), by letting t → ∞ we obtain”
A bifurcation result for non-local fractional equations (2015, background): “In [4] [7] [8] [11] [19] [20] [21] [23] [24] [25] [26] [27] [30] (see also the references therein) the authors studied non-local fractional Laplacian equations with superlinear and subcritical or critical nonlinearities, while in [12] [13] the asymptotically linear case was exploited.”
Ground state solutions of scalar field fractional Schrödinger equations (2015, methodology): “Remark 7 For completenesswe justmention here that, by using a variational approach similar to the one adopted here, in [36–38] the authors proved the existence and the multiplicity of weak solutions for nonlocal problems involving regional fractional Laplacian operators in a suitable abstract settin”
The Nehari manifold for a singular elliptic equation involving the fractional Laplace operator (2016, background): “integral fractional Laplace operator has been investigated in quite a large number of papers, such as [7, 9, 18, 19] and references therein.”
Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian (2012, background): “This fractional Sobolev exponent also plays a role for the nonlinear analysis methods for equations in bounded domains; see [17].”
On Fractional Laplacians (2013, background): “In this case both the operators (−∆Ω) s N and (−∆Ω) s D were considered in many articles on semilinear equations, see for instance [2, 9, 11, 12, 13, 14, 17], and compared in [8], [15].”
Higher nonlocal problems with bounded potential (2014, background): “For further details on the fractional Sobolev spaces we refer to [7] and to the references therein, while for other details on X and X0 we refer to [13], where these functional spaces were introduced, and also to [12, 14, 15, 16], where various properties of these spaces were proved.”
Three Weak Solutions for Nonlocal Fractional Equations (2014, neutral): “For further details on the fractional Sobolev spaces we refer to [1] and to the references therein, while for other details on X and X0 we refer to [12], where these functional spaces were introduced, and also to [11, 13, 14, 15], where various properties of these spaces were proved.”