Mountain Pass solutions for non-local elliptic operators

By Raffaella Servadei, Enrico Valdinoci. 2012

740 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/3540ceb33772f79fc7ea72badb78c5aedd980724

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In Applied Mathematics

In Computational Theory and Mathematics

In Numerical Analysis

Citations in the corpus

Radial symmetry of positive solutions to equations involving the fractional laplacian (2013, neutral): “Existence and related results were studied by Cabré and Tan [4], Dipierro, Palatucci and Valdinoci [12], Felmer, Quaas and Tan [13], and Servadei and Valdinoci [24].”
The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary (2012, background): “Other works treating the fractional Dirichlet semilinear problem, which deal mainly with existence of solutions and symmetry properties, are [22, 23, 12, 1].”
Saddle point solutions for non-local elliptic operators (2012, methodology): “It is immediate to observe that bounded and Lipschitz functions belong to X (see [7, 8] for further details on space X).”
Getting Acquainted with the Fractional Laplacian (2017, background): “[85]) and a viscosity solution approach is often extremely appropriate to construct general regularity theories (see e.”
BIFURCATION AND MULTIPLICITY RESULTS FOR CRITICAL NONLOCAL FRACTIONAL LAPLACIAN PROBLEMS (2016, background): “Q |g(x)− g(y)|(2)K(x− y)dx dy )1/2 , where Q := (Rn × Rn) \ O and O = (CΩ) × (CΩ) ⊂ Rn × Rn (see, for instance, [27] for a proof).”
On doubly nonlocal fractional elliptic equations (2015, methodology): “Technically, this approach is realizable by checking that the associated energy functional satisfies the usual compactness Palais-Smale condition in a suitable variational setting developed by Servadei and Valdinoci (see [30]).”
On critical systems involving fractional Laplacian (2017, positive): “Preliminaries We need the following results, which have been proved in [9].”
The Nehari manifold for a singular elliptic equation involving the fractional Laplace operator (2016, background): “‖) is a Hilbert space and the embedding E ↪→ L2(Ω) is continuous (for detail see [18]).”
Elliptic problems involving the fractional Laplacian in RN (2013, methodology): “A mountain pass theorem and applications to Dirichlet problems in bounded domains involving non-local integro-differential operators of fractional Laplacian type are given in [29].”
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”
WEAK AND VISCOSITY SOLUTIONS OF THE FRACTIONAL LAPLACE EQUATION (2014, neutral): “, [12, 14] and references therein).”