Positive solutions of nonlinear problems involving the square root of the Laplacian

By X. Cabré, Jinggang Tan. 2009

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Abstract We consider nonlinear elliptic problems involving a nonlocal operator: the square root of the Laplacian in a bounded domain with zero Dirichlet boundary conditions. For positive solutions to problems with power nonlinearities, we establish existence and regularity results, as well as a priori estimates of Gidas–Spruck type. In addition, among other results, we prove a symmetry theorem of Gidas–Ni–Nirenberg type.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/1d93e081668037b044d142ada520bd9297db2238

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Regularity of Radial Extremal Solutions for Some Non-Local Semilinear Equations (2010, result): “Cabré and Tan [10] obtained several results in analogy with the classical Lane-Emden problem − u = up, posed on bounded domains and entire space, related to the role of the critical exponent.”
Monotonicity and nonexistence results for some fractional elliptic problems in the half-space (2013, methodology): “ assumptions. This topic will be considered by the authors in a future work. The combination of Liouville type results with rescaling arguments has already been applied successfully by Cabr´e and Tan [13] for nonlinear boundary value problems involving the spectral theoretic square root of the Diric”
The Nehari manifold for a singular elliptic equation involving the fractional Laplace operator (2016, background): “Some other results dealing with the existence of solutions concerning Dirichlet problem involving the spectral fractional laplacian has been treated in [4, 6, 21] and references therein.”
Indefinite fractional elliptic problem and Liouville theorems (2014, methodology): “) α 2 u(x), where the constant kα= 21−αΓ(1− 1 2 α) Γ(1 2 α) . To deﬁne the fractional Laplacian in a bounded domain, the idea is to use the CaﬀarelliSilvestre’s extension in a cylindrical domain. See [CT] for the case α= 1 and [BCDS] for its generalization to 0 <α<2. Let {λk,φk}∞ k=1 be the ei”
DENSITY PROPERTIES FOR FRACTIONAL SOBOLEV SPACES (2015, background): “Recently, in the literature a deep interest was shown for nonlocal operators, thanks to their intriguing analytical structure and in view of several applications in a wide range of contexts, such as the thin obstacle problem, optimization, finance, phase transitions, stratified materials, anomalous ”
The Pohozaev Identity for the Fractional Laplacian (2012, background): “For such extension, some Pohozaev type identities are proved in [4, 5, 6].”
Semilinear elliptic equations for the fractional Laplacian with Hardy potential (2011, methodology): “13), it was shown in [11] and [14] that”
ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR NONLINEAR ELLIPTIC PROBLEMS WITH THE FRACTIONAL LAPLACIAN (2013, methodology): “By the celebrated results of Caffarelli-Silvestre [7] (for Rn) and Cabré-Tan [6] (for bounded domains, see also [38, 8, 2, 40]), if we set U ∈ Hs0,L(C) (or Ds(Rn+1+ )) as a unique solution of the equation div(t1−2s∇U) = 0 in C, U = 0 on ∂LC, U(x, 0) = u(x) for x ∈ Ω, (2.4) for some fixed functio”
The Brezis–Nirenberg type problem involving the square root of the Laplacian (2011, methodology): “Proposition 2.8. [ 7 ] Letf’kg be an orthonormal basis of L2() giving an spec-”
A concave—convex elliptic problem involving the fractional Laplacian (2010, background): “To define the fractional Laplacian in a bounded domain we follow [14], see also [36].”
Recent progress in the theory of Nonlinear Diffusion with Fractional Laplacian Operators (2014, methodology): “[48] R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC, 2004.”
On some critical problems for the fractional Laplacian operator (2011, background): “arelli and Silvestre [13], several authors have considered an equivalent deﬁnition of the operator (−∆)α/2 in a bounded domain with zero Dirichlet boundary data by means of an auxiliary variable, see [5, 11, 12, 14, 22]. Associated to the bounded domain Ω, let us consider the cylinder CΩ = Ω× (0,∞) ”
Equations involving fractional Laplacian operator: Compactness and application (2015, methodology): “onents, though the methods used in [12, 13, 26] to obtain the estimates are diﬀerent from those in [14]. Problems with the fractional Laplacian have been extensively studied recently. See for example [3, 4, 6, 7, 8, 10, 11, 16, 20, 22, 23, 24]. In particular, the Br´ezis-Nirenberg type problem was d”