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November  2019, 39(11): 6747-6760. doi: 10.3934/dcds.2019293

The jumping problem for nonlocal singular problems

Annamaria Canino , Luigi Montoro and  Berardino Sciunzi ,

Dipartimento di Matematica e Informatica, Ponte Pietro Bucci 31B, I-87036 Arcavacata di Rende, Cosenza, Italy

*Corresponding author: Berardino Sciunzi

Received  April 2019 Revised  June 2019 Published  August 2019

Fund Project: The authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

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.

Keywords: Fractional Laplacian, singular problems, jumping problems.
Mathematics Subject Classification: Primary: 35S05, 35S15, 35R11; Secondary: 60G22.
Citation: Annamaria Canino, Luigi Montoro, Berardino Sciunzi. The jumping problem for nonlocal singular problems. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6747-6760. doi: 10.3934/dcds.2019293
References:
[1]

H. W. Alt and D. Phillips, A free boundary problem for semilinear elliptic equations, J. Reine Angew. Math., 368 (1986), 63-107.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

A. Canino, On a jumping problem for quasilinear elliptic equations, Math. Z., 226 (1997), 193-210.  doi: 10.1007/PL00004336.  Google Scholar

[7]

A. Canino and M. Degiovanni, A variational approach to a class of singular semilinear elliptic equations, J. Convex Anal., 11 (2004), 147-162.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

A. Canino, L. Montoro and B. Sciunzi, A variational approach to nonlocal singular problems, preprint, arXiv: 1806.05670. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

G. Franzina and G. Palatucci, Fractional $p$-eigenvalues, Riv. Math. Univ. Parma (N.S.), 5 (2014), 373-386.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[19]

K. M. Hui, Global and touchdown behaviour of the generalized MEMS device equation, Adv. Math. Sci. Appl., 19 (2009), 347-370.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

E. Lindgren and P. Lindqvist, Fractional eigenvalues, Calc. Var. Partial Differential Equations, 49 (2014), 795–826. doi: 10.1007/s00526-013-0600-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. 1970.  Google Scholar

[27]

C. A. Stuart, Existence and approximation of solutions of non-linear elliptic equations, Math. Z., 147 (1976), 53-63.  doi: 10.1007/BF01214274.  Google Scholar

[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.  Google Scholar

[29]

E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl. SeMA, 49 (2009), 33-44.   Google Scholar

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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

A. Canino, On a jumping problem for quasilinear elliptic equations, Math. Z., 226 (1997), 193-210.  doi: 10.1007/PL00004336.  Google Scholar

[7]

A. Canino and M. Degiovanni, A variational approach to a class of singular semilinear elliptic equations, J. Convex Anal., 11 (2004), 147-162.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

A. Canino, L. Montoro and B. Sciunzi, A variational approach to nonlocal singular problems, preprint, arXiv: 1806.05670. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

G. Franzina and G. Palatucci, Fractional $p$-eigenvalues, Riv. Math. Univ. Parma (N.S.), 5 (2014), 373-386.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[19]

K. M. Hui, Global and touchdown behaviour of the generalized MEMS device equation, Adv. Math. Sci. Appl., 19 (2009), 347-370.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

E. Lindgren and P. Lindqvist, Fractional eigenvalues, Calc. Var. Partial Differential Equations, 49 (2014), 795–826. doi: 10.1007/s00526-013-0600-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. 1970.  Google Scholar

[27]

C. A. Stuart, Existence and approximation of solutions of non-linear elliptic equations, Math. Z., 147 (1976), 53-63.  doi: 10.1007/BF01214274.  Google Scholar

[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.  Google Scholar

[29]

E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl. SeMA, 49 (2009), 33-44.   Google Scholar

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