-
Previous Article
On the uniqueness of bound state solutions of a semilinear equation with weights
- DCDS Home
- This Issue
-
Next Article
Global well-posedness to the cauchy problem of two-dimensional density-dependent boussinesq equations with large initial data and vacuum
The jumping problem for nonlocal singular problems
Dipartimento di Matematica e Informatica, Ponte Pietro Bucci 31B, I-87036 Arcavacata di Rende, Cosenza, Italy |
We consider a jumping problem for nonlocal singular problems. We apply a recent variational approach for nonlocal singular problem, together with a minimax method in the framework of nonsmooth critical point theory.
References:
[1] |
H. W. Alt and D. Phillips,
A free boundary problem for semilinear elliptic equations, J. Reine Angew. Math., 368 (1986), 63-107.
|
[2] |
A. Ambrosetti and P. H. Rabinowitz,
Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.
doi: 10.1016/0022-1236(73)90051-7. |
[3] |
B. Barrios, I. De Bonis, M. Medina and I. Peral,
Semilinear problems for the fractional laplacian with a singular nonlinearity, Open Math., 13 (2015), 390-407.
doi: 10.1515/math-2015-0038. |
[4] |
L. Boccardo and L. Orsina,
Semilinear elliptic equations with singular nonlinearities, Calc. Var. Partial Differential Equations, 37 (2010), 363-380.
doi: 10.1007/s00526-009-0266-x. |
[5] |
A. Canino,
Minimax methods for singular elliptic equations with an application to a jumping problem, J. Differential Equations, 221 (2006), 210-223.
doi: 10.1016/j.jde.2005.01.015. |
[6] |
A. Canino,
On a jumping problem for quasilinear elliptic equations, Math. Z., 226 (1997), 193-210.
doi: 10.1007/PL00004336. |
[7] |
A. Canino and M. Degiovanni,
A variational approach to a class of singular semilinear elliptic equations, J. Convex Anal., 11 (2004), 147-162.
|
[8] |
A. Canino, M. Grandinetti and B. Sciunzi,
A jumping problem for some singular semilinear elliptic equations, Adv. Nonlinear Stud., 14 (2014), 1037-1054.
doi: 10.1515/ans-2014-0412. |
[9] |
A. Canino, M. Grandinetti and B. Sciunzi,
Symmetry of solutions of some semilinear elliptic equations with singular nonlinearities, J. Differential Equations, 255 (2013), 4437-4447.
doi: 10.1016/j.jde.2013.08.014. |
[10] |
A. Canino, L. Montoro and B. Sciunzi, A variational approach to nonlocal singular problems, preprint, arXiv: 1806.05670. |
[11] |
A. Canino, L. Montoro and B. Sciunzi,
The moving plane method for singular semilinear elliptic problems, Nonlinear Anal., 156 (2017), 61-69.
doi: 10.1016/j.na.2017.02.009. |
[12] |
A. Canino, L. Montoro, B. Sciunzi and M. Squassina,
Nonlocal problems with singular nonlinearity, Bull. Sci. Math., 141 (2017), 223-250.
doi: 10.1016/j.bulsci.2017.01.002. |
[13] |
M. G. Crandall, P. H. Rabinowitz and L. Tartar,
On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations, 2 (1977), 193-222.
doi: 10.1080/03605307708820029. |
[14] |
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. |
[15] |
G. Franzina and G. Palatucci,
Fractional $p$-eigenvalues, Riv. Math. Univ. Parma (N.S.), 5 (2014), 373-386.
|
[16] |
J. A. Gatica, V. Oliker and P. Waltman,
Singular nonlinear boundary value problems for second-order ordinary differential equations, J. Differential Equations, 79 (1989), 62-78.
doi: 10.1016/0022-0396(89)90113-7. |
[17] |
A. Groli,
Jumping problems for quasilinear elliptic variational inequalities, NoDEA Nonlinear Differential Equations Appl., 9 (2002), 117-147.
doi: 10.1007/s00030-002-8121-1. |
[18] |
N. Hirano, C. Saccon and N. Shioji,
Multiple existence of positive solutions for singular elliptic problems with concave ad convex nonlinearities, Adv. Differential Equations, 9 (2004), 197-220.
|
[19] |
K. M. Hui,
Global and touchdown behaviour of the generalized MEMS device equation, Adv. Math. Sci. Appl., 19 (2009), 347-370.
|
[20] |
B. Kawohl, On a class of singular elliptic equations, in Progress in partial differential equations: Elliptic and parabolic problems (Pont-à-Mousson, 1991), Pitman Res. Notes Math. Ser., 266 (1992), Longman Sci. Tech., Harlow, 156–163. |
[21] |
A. V. Lair and A. W. Shaker,
Classical and weak solutionsof a singular semilinear elliptic problem, J. Math. Anal. Appl., 211 (1977), 371-385.
doi: 10.1006/jmaa.1997.5470. |
[22] |
A. C. Lazer and P. J. McKenna,
On a singular nonlinear elliptic boundary-value problem, Proc. Amer. Math. Soc., 111 (1991), 721-730.
doi: 10.1090/S0002-9939-1991-1037213-9. |
[23] |
E. Lindgren and P. Lindqvist, Fractional eigenvalues, Calc. Var. Partial Differential Equations, 49 (2014), 795–826.
doi: 10.1007/s00526-013-0600-1. |
[24] |
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. |
[25] |
R. Servadei and E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat., 58 (2014), 133–154.
doi: 10.5565/PUBLMAT_58114_06. |
[26] |
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. 1970. |
[27] |
C. A. Stuart,
Existence and approximation of solutions of non-linear elliptic equations, Math. Z., 147 (1976), 53-63.
doi: 10.1007/BF01214274. |
[28] |
A. Szulkin,
Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 77-109.
doi: 10.1016/S0294-1449(16)30389-4. |
[29] |
E. Valdinoci,
From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl. SeMA, 49 (2009), 33-44.
|
show all references
References:
[1] |
H. W. Alt and D. Phillips,
A free boundary problem for semilinear elliptic equations, J. Reine Angew. Math., 368 (1986), 63-107.
|
[2] |
A. Ambrosetti and P. H. Rabinowitz,
Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.
doi: 10.1016/0022-1236(73)90051-7. |
[3] |
B. Barrios, I. De Bonis, M. Medina and I. Peral,
Semilinear problems for the fractional laplacian with a singular nonlinearity, Open Math., 13 (2015), 390-407.
doi: 10.1515/math-2015-0038. |
[4] |
L. Boccardo and L. Orsina,
Semilinear elliptic equations with singular nonlinearities, Calc. Var. Partial Differential Equations, 37 (2010), 363-380.
doi: 10.1007/s00526-009-0266-x. |
[5] |
A. Canino,
Minimax methods for singular elliptic equations with an application to a jumping problem, J. Differential Equations, 221 (2006), 210-223.
doi: 10.1016/j.jde.2005.01.015. |
[6] |
A. Canino,
On a jumping problem for quasilinear elliptic equations, Math. Z., 226 (1997), 193-210.
doi: 10.1007/PL00004336. |
[7] |
A. Canino and M. Degiovanni,
A variational approach to a class of singular semilinear elliptic equations, J. Convex Anal., 11 (2004), 147-162.
|
[8] |
A. Canino, M. Grandinetti and B. Sciunzi,
A jumping problem for some singular semilinear elliptic equations, Adv. Nonlinear Stud., 14 (2014), 1037-1054.
doi: 10.1515/ans-2014-0412. |
[9] |
A. Canino, M. Grandinetti and B. Sciunzi,
Symmetry of solutions of some semilinear elliptic equations with singular nonlinearities, J. Differential Equations, 255 (2013), 4437-4447.
doi: 10.1016/j.jde.2013.08.014. |
[10] |
A. Canino, L. Montoro and B. Sciunzi, A variational approach to nonlocal singular problems, preprint, arXiv: 1806.05670. |
[11] |
A. Canino, L. Montoro and B. Sciunzi,
The moving plane method for singular semilinear elliptic problems, Nonlinear Anal., 156 (2017), 61-69.
doi: 10.1016/j.na.2017.02.009. |
[12] |
A. Canino, L. Montoro, B. Sciunzi and M. Squassina,
Nonlocal problems with singular nonlinearity, Bull. Sci. Math., 141 (2017), 223-250.
doi: 10.1016/j.bulsci.2017.01.002. |
[13] |
M. G. Crandall, P. H. Rabinowitz and L. Tartar,
On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations, 2 (1977), 193-222.
doi: 10.1080/03605307708820029. |
[14] |
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. |
[15] |
G. Franzina and G. Palatucci,
Fractional $p$-eigenvalues, Riv. Math. Univ. Parma (N.S.), 5 (2014), 373-386.
|
[16] |
J. A. Gatica, V. Oliker and P. Waltman,
Singular nonlinear boundary value problems for second-order ordinary differential equations, J. Differential Equations, 79 (1989), 62-78.
doi: 10.1016/0022-0396(89)90113-7. |
[17] |
A. Groli,
Jumping problems for quasilinear elliptic variational inequalities, NoDEA Nonlinear Differential Equations Appl., 9 (2002), 117-147.
doi: 10.1007/s00030-002-8121-1. |
[18] |
N. Hirano, C. Saccon and N. Shioji,
Multiple existence of positive solutions for singular elliptic problems with concave ad convex nonlinearities, Adv. Differential Equations, 9 (2004), 197-220.
|
[19] |
K. M. Hui,
Global and touchdown behaviour of the generalized MEMS device equation, Adv. Math. Sci. Appl., 19 (2009), 347-370.
|
[20] |
B. Kawohl, On a class of singular elliptic equations, in Progress in partial differential equations: Elliptic and parabolic problems (Pont-à-Mousson, 1991), Pitman Res. Notes Math. Ser., 266 (1992), Longman Sci. Tech., Harlow, 156–163. |
[21] |
A. V. Lair and A. W. Shaker,
Classical and weak solutionsof a singular semilinear elliptic problem, J. Math. Anal. Appl., 211 (1977), 371-385.
doi: 10.1006/jmaa.1997.5470. |
[22] |
A. C. Lazer and P. J. McKenna,
On a singular nonlinear elliptic boundary-value problem, Proc. Amer. Math. Soc., 111 (1991), 721-730.
doi: 10.1090/S0002-9939-1991-1037213-9. |
[23] |
E. Lindgren and P. Lindqvist, Fractional eigenvalues, Calc. Var. Partial Differential Equations, 49 (2014), 795–826.
doi: 10.1007/s00526-013-0600-1. |
[24] |
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. |
[25] |
R. Servadei and E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat., 58 (2014), 133–154.
doi: 10.5565/PUBLMAT_58114_06. |
[26] |
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. 1970. |
[27] |
C. A. Stuart,
Existence and approximation of solutions of non-linear elliptic equations, Math. Z., 147 (1976), 53-63.
doi: 10.1007/BF01214274. |
[28] |
A. Szulkin,
Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 77-109.
doi: 10.1016/S0294-1449(16)30389-4. |
[29] |
E. Valdinoci,
From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl. SeMA, 49 (2009), 33-44.
|
[1] |
Yutong Chen, Jiabao Su. Resonant problems for fractional Laplacian. Communications on Pure and Applied Analysis, 2017, 16 (1) : 163-188. doi: 10.3934/cpaa.2017008 |
[2] |
Tuhina Mukherjee, Patrizia Pucci, Mingqi Xiang. Combined effects of singular and exponential nonlinearities in fractional Kirchhoff problems. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 163-187. doi: 10.3934/dcds.2021111 |
[3] |
Alberto Cabada, J. Ángel Cid. Heteroclinic solutions for non-autonomous boundary value problems with singular $\Phi$-Laplacian operators. Conference Publications, 2009, 2009 (Special) : 118-122. doi: 10.3934/proc.2009.2009.118 |
[4] |
Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3497-3528. doi: 10.3934/dcdss.2020442 |
[5] |
Mingqi Xiang, Binlin Zhang, Die Hu. Kirchhoff-type differential inclusion problems involving the fractional Laplacian and strong damping. Electronic Research Archive, 2020, 28 (2) : 651-669. doi: 10.3934/era.2020034 |
[6] |
Yutong Chen, Jiabao Su. Nontrivial solutions for the fractional Laplacian problems without asymptotic limits near both infinity and zero. Discrete and Continuous Dynamical Systems - S, 2021, 14 (6) : 1837-1855. doi: 10.3934/dcdss.2021007 |
[7] |
Simone Creo, Maria Rosaria Lancia, Paola Vernole. Transmission problems for the fractional $ p $-Laplacian across fractal interfaces. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022047 |
[8] |
Mohammed AL Horani, Mauro Fabrizio, Angelo Favini, Hiroki Tanabe. Fractional Cauchy problems and applications. Discrete and Continuous Dynamical Systems - S, 2020, 13 (8) : 2259-2270. doi: 10.3934/dcdss.2020187 |
[9] |
Maria Fărcăşeanu, Mihai Mihăilescu, Denisa Stancu-Dumitru. Perturbed fractional eigenvalue problems. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6243-6255. doi: 10.3934/dcds.2017270 |
[10] |
Paulo Cesar Carrião, R. Demarque, Olímpio H. Miyagaki. Nonlinear Biharmonic Problems with Singular Potentials. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2141-2154. doi: 10.3934/cpaa.2014.13.2141 |
[11] |
Qiang Lin, Xueteng Tian, Runzhang Xu, Meina Zhang. Blow up and blow up time for degenerate Kirchhoff-type wave problems involving the fractional Laplacian with arbitrary positive initial energy. Discrete and Continuous Dynamical Systems - S, 2020, 13 (7) : 2095-2107. doi: 10.3934/dcdss.2020160 |
[12] |
Jacques Giacomoni, Tuhina Mukherjee, Konijeti Sreenadh. Existence and stabilization results for a singular parabolic equation involving the fractional Laplacian. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 311-337. doi: 10.3934/dcdss.2019022 |
[13] |
Genni Fragnelli, Dimitri Mugnai, Nikolaos S. Papageorgiou. Robin problems for the p-Laplacian with gradient dependence. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 287-295. doi: 10.3934/dcdss.2019020 |
[14] |
Friedemann Brock, Leonelo Iturriaga, Justino Sánchez, Pedro Ubilla. Existence of positive solutions for $p$--Laplacian problems with weights. Communications on Pure and Applied Analysis, 2006, 5 (4) : 941-952. doi: 10.3934/cpaa.2006.5.941 |
[15] |
Luis Caffarelli, Antoine Mellet. Random homogenization of fractional obstacle problems. Networks and Heterogeneous Media, 2008, 3 (3) : 523-554. doi: 10.3934/nhm.2008.3.523 |
[16] |
Ida De Bonis, Daniela Giachetti. Singular parabolic problems with possibly changing sign data. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 2047-2064. doi: 10.3934/dcdsb.2014.19.2047 |
[17] |
M. Delgado-Téllez, Alberto Ibort. On the geometry and topology of singular optimal control problems and their solutions. Conference Publications, 2003, 2003 (Special) : 223-233. doi: 10.3934/proc.2003.2003.223 |
[18] |
Prashanta Garain, Tuhina Mukherjee. Quasilinear nonlocal elliptic problems with variable singular exponent. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5059-5075. doi: 10.3934/cpaa.2020226 |
[19] |
Eunkyoung Ko, Eun Kyoung Lee, R. Shivaji. Multiplicity results for classes of singular problems on an exterior domain. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5153-5166. doi: 10.3934/dcds.2013.33.5153 |
[20] |
Maria Assunta Pozio, Fabio Punzo, Alberto Tesei. Uniqueness and nonuniqueness of solutions to parabolic problems with singular coefficients. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 891-916. doi: 10.3934/dcds.2011.30.891 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]