Recent Progresses in the Theory of Nonlinear Nonlocal Problems

Sunra Mosconi, Marco Squassina

Abstract


We overview some recent existence and regularity results in the theory of nonlocal nonlinear problems driven by the fractional p-Laplacian.


Keywords


Fractional p-Laplacian; Existence results; Regularity results

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References


T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Dierential Geometry 11 (1976), 573-598.

G. Arioli, F. Gazzola, Some results on p-Laplace equations with a critical growth term, Differential Integral Equations 11 (1998), 311-326.

L. Brasco, E. Lindgren, Higher Sobolev regularity for the fractional p-Laplace equation in the superquadratic case, Adv. Math, 304 (2017), 300-354.

L. Brasco, S. Mosconi, M. Squassina, Optimal decay of extremals for the fractional Sobolev inequality, Calc. Var. Partial Differential Equations 55 (2016), 55:23.

L. Brasco, E. Parini, The second eigenvalue of the fractional p-Laplacian, Adv. Calc. Var. 9 (2016), 323-355.

L. Brasco, F. Santambrogio, A sharp estimate à la Calderón-Zygmund for the p-laplacian, Comm. Contemp. Math., to appear

H. Brézis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 437-477.

L.A. Caffarelli, Nonlocal equations, drifts and games, Nonlinear Partial Differential Equations, Abel Symposia 7 (2012) 37-52.

L. Caffarelli, L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl.Math. 62 (2009), 597-638.

L. Caffarelli, L. Silvestre, Regularity results for nonlocal equations by approximation, Arch. Ration. Mech. Anal. 200 (2011), 59-88.

A. Capozzi, D. Fortunato, G. Palmieri, An existence result for nonlinear elliptic problems involving critical

Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), 463-470.

X. Chang, Z-Q. Wang, Ground state of scalar eld equations involving a fractional laplacian with general

nonlinearity, Nonlinearity 26 (2013), 479-494.

W. Chen, C. Li, B. Ou, Classication of solutions for an integral equation, Comm. Pure Appl. Math. 59 (2006),

-343.

W. Chen, C. Li, B. Ou, Qualitative properties of solutions for an integral equation Discrete Contin. Dyn. Syst.

(2005), 347-354.

D. Cordero-Erausquin, B. Nazaret, C. Villani, A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities, Adv. Math. 182 (2004), 307-332.

A. Di Castro, T. Kuusi, G. Palatucci, Local behavior of fractional p-minimizers, Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), 1279-1299.

L. Damascelli, S. Merchán, L. Montoro, B. Sciunzi, Radial symmetry and applications for a problem involving the -∆p(.) operator and critical nonlinerity in RN, Adv. Math. 265 (2014), 313-335.

M. Degiovanni, S. Lancelotti, Linking solutions for p-Laplace equations with nonlinearity at critical growth, J.Funct. Anal. 256 (2009), 3643-3659.

M. Degiovanni, A. Musesti, M. Squassina, On the regularity of solutions in the Pucci-Serrin identity, Calc.Var. Partial Differential Equations 18 (2003), 317-334.

W.-Y. Ding, On a conformally invariant elliptic equation on RN, Comm. Math. Phys. 107 (1986), 331-335.

A. Farina, On the classication of solutions of the Lane-Emden equation on unbounded domains of RN, J.Math. Pures Appl. 87 (2007), 537-561.

J. P. García Azorero, I. Peral Alonso, Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term, Trans. Amer. Math. Soc. 323 (1991), 877-895.

G. Grubb, Local and nonlocal boundary conditions for μ-transmission and fractional elliptic pseudo-differential operators, Anal. PDE 7 (2014), 1649-1682.

G. Grubb, Fractional Laplacians on domains, a development of Hörmander's theory of μ-transmission pseudo-differential operators, Adv. Math. 268 (2015), 478-528.

M. Guedda, L. Veron, Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal. 13 (1989), 879-902.

A. Iannizzotto, S. Liu, K. Perera, M. Squassina, Existence results for fractional p-Laplacian problems via Morse theory, Adv. Calc. Var. 9 (2016), 101-125.

A. Iannizzotto, S. Mosconi, M. Squassina, Hs versus C0-weighted minimizers, NoDEA Nonlinear Differential Equations Appl. 22 (2015), 477-497.

A. Iannizzotto, S. Mosconi, M. Squassina, Global Hölder regularity for the fractional p-Laplacian, Rev. Mat. Iberoamer. 32 (2016), 1355-1394.

A. Iannizzotto, M. Squassina, Weyl-type laws for fractional p-eigenvalue problems, Asymptotic Anal. 88 2014),233-245.

J. Korvenpää, T. Kuusi, G. Palatucci, The obstacle problem for nonlinear integro-differential operators, Calc. Var. Partial Differential Equations 55 (2016) 3:63.

I. Kupka, Counterexample to the Morse-Sard Theorem in the Case of Infinite-Dimensional Manifolds Proc. Amer. Math. Soc. 16 (1965), 954-957.

T. Kuusi, G. Mingione, Y. Sire, Nonlocal equations with measure data, Comm. Math. Phys. 337 (2015), 1317-1368.

E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalies, Ann. Math. 118 (1983), 349-374.

E. Carlen, M. Loss, Extremals of functionals with competing symmetries, J. Funct. Anal. 88 (1990), 437-456.

E. Lindgren, P. Lindqvist, Fractional eigenvalues, Calc. Var. Partial Differential Equations 49 (2014), 795-826.

S. Marano, S. Mosconi, Asymptotics for optimizers of the fractional Hardy-Sobolev inequality, preprint arXiv http://arxiv.org/abs/1609.00179.

S. Mosconi, K. Perera, M. Squassina, Y. Yang, The Brezis-Nirenberg problem for the fractional p-Laplacian, Calc. Var. Partial Differential Equations 55 (2016), 55:105.

S. Mosconi, N. Shioji, M. Squassina, Nonlocal problems at critical growth in contractible domains, Asymptotic Anal. 95 (2015), 79-100.

S. Mosconi, M. Squassina, Nonlocal problems at nearly critical growth, preprint, Nonlinear Anal. 136 (2016), 84-101.

A. Norton, A critical set with nonnull image has large Hausdorff dimension, Trans. Amer. Math. Soc. 296 (1986), 367-376.

A. Norton, Functions not constant on fractal quasi-arcs of critical points, Proc. Amer. Math. Soc. 106 (1989), 397-405.

K. Perera, M. Squassina, Y. Yang, A note on the Dancer-Fucik spectra of the fractional p-Laplacian and Laplacian operators, Adv. Nonlinear Anal. 8 (2015), 13-23.

P. Pucci, J. Serrin, A general variational identity, Indiana Univ. Math. J. 35 (1986), 681-703.

X. Ros-Oton, J. Serra, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl. 101 (2014), 275-302.

X. Ros-Oton, J. Serra, The Pohŏzaev identity for the fractional Laplacian, Arch. Rat. Mech. Anal. 213 (2014), 587-628.

X. Ros-Oton, J. Serra, Nonexistence Results for Nonlocal Equations with Critical and Supercritical Nonlinearities Comm. Partial Differential Equations 40 (2015), 115-133.

X. Ros-Oton, J. Serra, Regularity theory for general stable operators, J. Differential Equations 260 (2016), 8675-8715.

X. Ros-Oton, J. Serra, Boundary regularity for fully nonlinear integro-differential equations, Duke Math. J. 165 (2016), 2079-2154.

X. Ros-Oton, J. Serra, Boundary regularity estimates for nonlocal elliptic equations in C1 and C1,α domains, preprint arXiv http://arxiv.org/abs/1512.07171.

J. Serra, Cσ+α regularity for concave nonlocal fully nonlinear elliptic equations with rough kernels, Calc. Var. Partial Differential Equations 54 (2015), 3571-3601.

B. Sciunzi, Classication of positive D1,p(RN)-solutions to the critical p-Laplace equation in RN, Adv. Math. 291 (2016), 12-23.

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. 4 (1976), 353-372.

J. Vetois, A priori estimates and application to the symmetry of solutions for critical p-Laplace equations, J. Differential Equations 260 (2016), 149-161.

H. Whitney, A function not constant on a connected set of critical points, Duke Math. J. 1 (1935), 514-517.




DOI: 10.6092/issn.2240-2829/6696

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