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Nonlinear Dirichlet problem for the nonlocal anisotropic operator $ L_K $
Department of Mathematics and Computer Science, University of Cagliari, Viale L. Merello 92, 09123 Cagliari, Italy |
In this paper we study an equation driven by a nonlocal anisotropic operator with homogeneous Dirichlet boundary conditions. We find at least three non trivial solutions: one positive, one negative and one of unknown sign, using variational methods and Morse theory. We present some results about regularity of solutions such as $ L^{\infty} $ bound and Hopf's lemma, for the latter we first consider a non negative nonlinearity and then a strictly negative one. Moreover, we prove that, for the corresponding functional, local minimizers with respect to a $ C^0 $-topology weighted with a suitable power of the distance from the boundary are actually local minimizers in the $ X(\Omega) $-topology.
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D. Applebaum,
Lévy processes-From probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347.
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[2] |
B. Barrios, E. Colorado, R. Servadei and F. Soria,
A critical fractional equation with concave-convex power nonlinearities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 875-900.
doi: 10.1016/j.anihpc.2014.04.003. |
[3] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.
doi: 10.1007/978-1-4612-0873-0. |
[4] |
H. Brezis and L. Nirenberg,
$H^1$ versus $C^1$ local minimizers, C. R. Acad. Sci. Paris Ser. I, 317 (1993), 465-472.
|
[5] |
C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Vol.20., Springer, Bologna, 2016.
doi: 10.1007/978-1-4612-0873-0. |
[6] |
L. Caffarelli, Non-local diffusions, drifts and games, in Nonlinear Partial Differential Equations, Springer, Berlin, Heidelberg, (2012), 37–52.
doi: 10.1007/978-3-642-25361-4_3. |
[7] |
F. Demengel, G. Demengel and R. Erné, Functional Spaces for the Theory of Elliptic Partial Differential Equations, Springer, London, 2012.
doi: 10.1007/978-1-4612-0873-0. |
[8] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[9] |
S. Dipierro, X. Ros-Oton and E. Valdinoci,
Nonlocal problems with Neumann boundary conditions, Rev. Mat. Iberoam., 33 (2017), 377-416.
doi: 10.4171/RMI/942. |
[10] |
F. G. Düzgün and A. Iannizzotto,
Three nontrivial solutions for nonlinear fractional Laplacian equations, Adv. Nonlinear Anal., 7 (2018), 211-226.
doi: 10.1515/anona-2016-0090. |
[11] |
A. Fiscella, R. Servadei and E. Valdinoci,
Density properties for fractional Sobolev spaces, Ann. Acad. Sci. Fenn. Math., 40 (2015), 235-253.
doi: 10.5186/aasfm.2015.4009. |
[12] |
A. Greco and R. Servadei,
Hopf's lemma and constrained radial symmetry for the fractional Laplacian, Math. Res. Lett., 23 (2016), 863-885.
doi: 10.4310/MRL.2016.v23.n3.a14. |
[13] |
A. Iannizzotto and M. Squassina,
1/2-Laplacian problems with exponential nonlinearity, J. Math. Anal. Appl., 414 (2014), 372-385.
doi: 10.1016/j.jmaa.2013.12.059. |
[14] |
A. Iannizzotto, S. Mosconi and M. Squassina,
$H^s$ versus $C^0$-weighted minimizers, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 477-497.
doi: 10.1007/s00030-014-0292-z. |
[15] |
A. Iannizzotto, S. Liu, K. Perera and M. Squassina,
Existence results for fractional p-Laplacian problems via Morse theory, Adv. Calc. Var., 9 (2016), 101-125.
doi: 10.1515/acv-2014-0024. |
[16] |
G. Molica Bisci, V. D. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Cambridge University Press, 2016.
doi: 10.1007/978-1-4612-0873-0. |
[17] |
D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, Springer, New York, 2014.
doi: 10.1007/978-1-4612-0873-0. |
[18] |
P. Pucci and J. Serrin,
A mountain pass theorem, J. Differential Equations, 60 (1985), 142-149.
doi: 10.1016/0022-0396(85)90125-1. |
[19] |
X. Ros-Oton,
Nonlocal elliptic equations in bounded domains: a survey, Publ. Mat., 60 (2016), 3-26.
|
[20] |
X. Ros-Oton and E. Valdinoci,
The Dirichlet problem for nonlocal operators with singular kernels: convex and nonconvex domains, Adv. Math., 288 (2016), 732-790.
doi: 10.1016/j.aim.2015.11.001. |
[21] |
R. Servadei and E. Valdinoci,
Mountain Pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898.
doi: 10.1016/j.jmaa.2011.12.032. |
[22] |
R. Servadei and E. Valdinoci,
Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.
doi: 10.3934/dcds.2013.33.2105. |
show all references
References:
[1] |
D. Applebaum,
Lévy processes-From probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347.
|
[2] |
B. Barrios, E. Colorado, R. Servadei and F. Soria,
A critical fractional equation with concave-convex power nonlinearities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 875-900.
doi: 10.1016/j.anihpc.2014.04.003. |
[3] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.
doi: 10.1007/978-1-4612-0873-0. |
[4] |
H. Brezis and L. Nirenberg,
$H^1$ versus $C^1$ local minimizers, C. R. Acad. Sci. Paris Ser. I, 317 (1993), 465-472.
|
[5] |
C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Vol.20., Springer, Bologna, 2016.
doi: 10.1007/978-1-4612-0873-0. |
[6] |
L. Caffarelli, Non-local diffusions, drifts and games, in Nonlinear Partial Differential Equations, Springer, Berlin, Heidelberg, (2012), 37–52.
doi: 10.1007/978-3-642-25361-4_3. |
[7] |
F. Demengel, G. Demengel and R. Erné, Functional Spaces for the Theory of Elliptic Partial Differential Equations, Springer, London, 2012.
doi: 10.1007/978-1-4612-0873-0. |
[8] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[9] |
S. Dipierro, X. Ros-Oton and E. Valdinoci,
Nonlocal problems with Neumann boundary conditions, Rev. Mat. Iberoam., 33 (2017), 377-416.
doi: 10.4171/RMI/942. |
[10] |
F. G. Düzgün and A. Iannizzotto,
Three nontrivial solutions for nonlinear fractional Laplacian equations, Adv. Nonlinear Anal., 7 (2018), 211-226.
doi: 10.1515/anona-2016-0090. |
[11] |
A. Fiscella, R. Servadei and E. Valdinoci,
Density properties for fractional Sobolev spaces, Ann. Acad. Sci. Fenn. Math., 40 (2015), 235-253.
doi: 10.5186/aasfm.2015.4009. |
[12] |
A. Greco and R. Servadei,
Hopf's lemma and constrained radial symmetry for the fractional Laplacian, Math. Res. Lett., 23 (2016), 863-885.
doi: 10.4310/MRL.2016.v23.n3.a14. |
[13] |
A. Iannizzotto and M. Squassina,
1/2-Laplacian problems with exponential nonlinearity, J. Math. Anal. Appl., 414 (2014), 372-385.
doi: 10.1016/j.jmaa.2013.12.059. |
[14] |
A. Iannizzotto, S. Mosconi and M. Squassina,
$H^s$ versus $C^0$-weighted minimizers, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 477-497.
doi: 10.1007/s00030-014-0292-z. |
[15] |
A. Iannizzotto, S. Liu, K. Perera and M. Squassina,
Existence results for fractional p-Laplacian problems via Morse theory, Adv. Calc. Var., 9 (2016), 101-125.
doi: 10.1515/acv-2014-0024. |
[16] |
G. Molica Bisci, V. D. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Cambridge University Press, 2016.
doi: 10.1007/978-1-4612-0873-0. |
[17] |
D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, Springer, New York, 2014.
doi: 10.1007/978-1-4612-0873-0. |
[18] |
P. Pucci and J. Serrin,
A mountain pass theorem, J. Differential Equations, 60 (1985), 142-149.
doi: 10.1016/0022-0396(85)90125-1. |
[19] |
X. Ros-Oton,
Nonlocal elliptic equations in bounded domains: a survey, Publ. Mat., 60 (2016), 3-26.
|
[20] |
X. Ros-Oton and E. Valdinoci,
The Dirichlet problem for nonlocal operators with singular kernels: convex and nonconvex domains, Adv. Math., 288 (2016), 732-790.
doi: 10.1016/j.aim.2015.11.001. |
[21] |
R. Servadei and E. Valdinoci,
Mountain Pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898.
doi: 10.1016/j.jmaa.2011.12.032. |
[22] |
R. Servadei and E. Valdinoci,
Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.
doi: 10.3934/dcds.2013.33.2105. |
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