-
Previous Article
Global attractors for a mixture problem in one dimensional solids with nonlinear damping and sources terms
- CPAA Home
- This Issue
-
Next Article
Second order non-autonomous lattice systems and their uniform attractors
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.
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. |
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. |
[1] |
Dorota Bors. Application of Mountain Pass Theorem to superlinear equations with fractional Laplacian controlled by distributed parameters and boundary data. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 29-43. doi: 10.3934/dcdsb.2018003 |
[2] |
Claudianor O. Alves, Giovany M. Figueiredo, Marcelo F. Furtado. Multiplicity of solutions for elliptic systems via local Mountain Pass method. Communications on Pure and Applied Analysis, 2009, 8 (6) : 1745-1758. doi: 10.3934/cpaa.2009.8.1745 |
[3] |
Raffaella Servadei, Enrico Valdinoci. Variational methods for non-local operators of elliptic type. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 2105-2137. doi: 10.3934/dcds.2013.33.2105 |
[4] |
Ian Schindler, Kyril Tintarev. Mountain pass solutions to semilinear problems with critical nonlinearity. Conference Publications, 2007, 2007 (Special) : 912-919. doi: 10.3934/proc.2007.2007.912 |
[5] |
Dmitry Glotov, P. J. McKenna. Numerical mountain pass solutions of Ginzburg-Landau type equations. Communications on Pure and Applied Analysis, 2008, 7 (6) : 1345-1359. doi: 10.3934/cpaa.2008.7.1345 |
[6] |
Christopher Grumiau, Marco Squassina, Christophe Troestler. On the Mountain-Pass algorithm for the quasi-linear Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1345-1360. doi: 10.3934/dcdsb.2013.18.1345 |
[7] |
Xinjing Wang. Liouville type theorem for Fractional Laplacian system. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5253-5268. doi: 10.3934/cpaa.2020236 |
[8] |
Lorenzo Brasco, Enea Parini, Marco Squassina. Stability of variational eigenvalues for the fractional $p-$Laplacian. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 1813-1845. doi: 10.3934/dcds.2016.36.1813 |
[9] |
Yixuan Wu, Yanzhi Zhang. Highly accurate operator factorization methods for the integral fractional Laplacian and its generalization. Discrete and Continuous Dynamical Systems - S, 2022, 15 (4) : 851-876. doi: 10.3934/dcdss.2022016 |
[10] |
Juan-Luis Vázquez. Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators. Discrete and Continuous Dynamical Systems - S, 2014, 7 (4) : 857-885. doi: 10.3934/dcdss.2014.7.857 |
[11] |
Siwei Duo, Hong Wang, Yanzhi Zhang. A comparative study on nonlocal diffusion operators related to the fractional Laplacian. Discrete and Continuous Dynamical Systems - B, 2019, 24 (1) : 231-256. doi: 10.3934/dcdsb.2018110 |
[12] |
Hantaek Bae. On the local and global existence of the Hall equations with fractional Laplacian and related equations. Networks and Heterogeneous Media, 2022 doi: 10.3934/nhm.2022021 |
[13] |
Shixiu Zheng, Zhilei Xu, Huan Yang, Jintao Song, Zhenkuan Pan. Comparisons of different methods for balanced data classification under the discrete non-local total variational framework. Mathematical Foundations of Computing, 2019, 2 (1) : 11-28. doi: 10.3934/mfc.2019002 |
[14] |
Maria Alessandra Ragusa, Atsushi Tachikawa. Estimates of the derivatives of minimizers of a special class of variational integrals. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1411-1425. doi: 10.3934/dcds.2011.31.1411 |
[15] |
Orlando Lopes. Uniqueness and radial symmetry of minimizers for a nonlocal variational problem. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2265-2282. doi: 10.3934/cpaa.2019102 |
[16] |
Yousef Alnafisah, Hamdy M. Ahmed. Neutral delay Hilfer fractional integrodifferential equations with fractional brownian motion. Evolution Equations and Control Theory, 2022, 11 (3) : 925-937. doi: 10.3934/eect.2021031 |
[17] |
Pavel Drábek, Stephen Robinson. Continua of local minimizers in a quasilinear model of phase transitions. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 163-172. doi: 10.3934/dcds.2013.33.163 |
[18] |
Augusto VisintiN. On the variational representation of monotone operators. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 909-918. doi: 10.3934/dcdss.2017046 |
[19] |
Annalisa Cesaroni, Matteo Novaga. Volume constrained minimizers of the fractional perimeter with a potential energy. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 715-727. doi: 10.3934/dcdss.2017036 |
[20] |
Henk Broer, Konstantinos Efstathiou, Olga Lukina. A geometric fractional monodromy theorem. Discrete and Continuous Dynamical Systems - S, 2010, 3 (4) : 517-532. doi: 10.3934/dcdss.2010.3.517 |
2021 Impact Factor: 1.273
Tools
Metrics
Other articles
by authors
[Back to Top]