Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian

By Patricio Felmer, Alexander Quaas, Jinggang Tan. Proceedings of the Royal Society of Edinburgh. Section A Mathematics, 2012

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Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5b1af4880a644f22793663e41dbc9485546228ba

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Citations in the corpus

Existence and symmetry of solutions for critical fractional Schrödinger equations with bounded potentials (2016, background): “Here we would like to point out some interesting models involving the fractional Laplacian, such as, the fractional Schrödinger equation (see [14, 15, 22, 23, 24]), the fractional Kirchhoff equation (see [16, 32, 33, 46, 47]), the fractional porous medium equation (see [9, 45]), the fractional Yamab”
Ground state solutions of scalar field fractional Schrödinger equations (2015, background): “Contrary to the classical Laplacian case that is widely investigated, the situation seems to be in a developing state when s ∈ (0, 1) and of an increasing interest (see, for instance, [19,21,24,26]).”
Multiplicity and concentration of positive solutions for the fractional Schrödinger–Poisson systems with critical growth (2017, neutral): “Observing that u ∈ L ∞ ( R 3 ), by the deﬁnition of φ u,t ( x ), there exists C > 0 such that Then it follows from Theorem 3.4 in [19] that there exists σ ∈ (0 , 1) such that u ∈ C 0 ,σ .”
Multiplicity results for the non-homogeneous fractional p-Kirchhoff equations with concave–convex nonlinearities (2015, background): “5) where the nonlinearity f satisfies some general conditions, for instance, see [28–30] for related discussions.”
Sharp threshold of blow-up and scattering for the fractional Hartree equation (2017, background): “The general fractional Laplacian was first proposed by Caffarelli and Silvestre in [4], and many researchers have studied the related time-independent Schrödinger equations with the fractional Laplacian (see[6, 11, 12, 16, 17, 32]).”
Semilinear elliptic equations for the fractional Laplacian with Hardy potential (2011, background): “We refer the reader to papers [10], [39], [40], [24], [4] (and the references there in) for a nice expository.”
Fractional double-phase patterns: concentration and multiplicity of solutions (2020, background): “Moreover, in order to show that the solutions of the modified problem are also solutions to (3), we can not adapt in our setting the arguments in [2,5,46], due to the nonlocal character of fractional p&q-Laplacian operators, and also fails the strategy used in [6] to study (4) based on some useful e”
Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum (2014, background): “(1-3) See, for instance, [Felmer et al. 2012] for the existence of such a solution and its basic properties.”
Existence and concentration of solution for a class of fractional elliptic equation in $\mathbb{R}^N$ via penalization method (2015, result): “22) Using some results found in [16], we know that”