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July  2019, 18(4): 1847-1867. doi: 10.3934/cpaa.2019086

Nonlinear Dirichlet problem for the nonlocal anisotropic operator $ L_K $

Silvia Frassu ,

Department of Mathematics and Computer Science, University of Cagliari, Viale L. Merello 92, 09123 Cagliari, Italy

* Corresponding author

Received  July 2018 Revised  September 2018 Published  January 2019

Fund Project: The author is member of GNAMPA (Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica 'Francesco Severi').

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.

Keywords: Integrodifferential operators, Variational methods, Fractional Laplacian, Local minimizers, Mountain Pass Theorem.
Mathematics Subject Classification: 35R09, 35R11, 47G20.
Citation: Silvia Frassu. Nonlinear Dirichlet problem for the nonlocal anisotropic operator $ L_K $. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1847-1867. doi: 10.3934/cpaa.2019086
References:
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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.  Google Scholar

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C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Vol.20., Springer, Bologna, 2016. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[19]

X. Ros-Oton, Nonlocal elliptic equations in bounded domains: a survey, Publ. Mat., 60 (2016), 3-26.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

D. Applebaum, Lévy processes-From probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347.   Google Scholar

[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.  Google Scholar

[3]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[4]

H. Brezis and L. Nirenberg, $H^1$ versus $C^1$ local minimizers, C. R. Acad. Sci. Paris Ser. I, 317 (1993), 465-472.   Google Scholar

[5]

C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Vol.20., Springer, Bologna, 2016. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

X. Ros-Oton, Nonlocal elliptic equations in bounded domains: a survey, Publ. Mat., 60 (2016), 3-26.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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