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June  2021, 14(6): 1837-1855. doi: 10.3934/dcdss.2021007

Nontrivial solutions for the fractional Laplacian problems without asymptotic limits near both infinity and zero

Yutong Chen and  Jiabao Su ,

School of Mathematical Sciences, Capital Normal University, Beijing 100048, China

* Corresponding author: Jiabao Su

Received  August 2020 Revised  November 2020 Published  June 2021 Early access  January 2021

Fund Project: Supported by NSFC(12001382, 11771302) and KZ202010028048

In this paper we obtain the existence of nontrivial solutions for the fractional Laplacian equations with the nonlinearity may fail to have asymptotic limits at zero and at infinity. We make use of a combination of homotopy invariance of critical groups and the topological version of linking methods.

Keywords: Fractional Laplacian, homotopy invariance, critical groups, Morse theory.
Mathematics Subject Classification: Primary: 35A15, 35A16, 35R09, 35R11, 58E05.
Citation: 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
References:
[1]

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.

[2]

M. Badiale and E. Serra, Semilinear Elliptic Equations for Beginners, Springer, Berlin, 2011. doi: 10.1007/978-0-85729-227-8.

[3]

B. Barrios, E. Colorado, A. de Pablo and U. Sánchez, On some critical problems for the fractional Laplacian operator, J. Differential Equations, 252 (2012), 6133-6162.  doi: 10.1016/j.jde.2012.02.023.

[4]

T. Bartsch and S. Li, Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlinear Anal., 28 (1997), 419-441.  doi: 10.1016/0362-546X(95)00167-T.

[5]

K.-C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser-Bosten, 1993. doi: 10.1007/978-1-4612-0385-8.

[6]

Y. Chen and J. Su, Resonant problems for fractional Laplacian, Commun. Pure Appl. Anal., 16 (2017), 163-187.  doi: 10.3934/cpaa.2017008.

[7]

Y. Chen and J. Su, Multiple solutions for the fractional Laplacian problems with different asymptotic limits near infinity, Appl. Math. Lett., 76 (2018), 60-65.  doi: 10.1016/j.aml.2017.07.012.

[8]

Y. Chen and J. Su, Bounded resonant problems driven by fractional Laplacian, Topol. Methods Nonlinear Anal., to appear.

[9]

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.

[10]

M. M. Fall and V. Felli, Unique continuation property and local asymptotics of solutions to fractional elliptic equations, Comm. Partial Differential Equations, 39 (2014), 354-397.  doi: 10.1080/03605302.2013.825918.

[11]

A. Fiscella, Saddle point solutions for nonlocal elliptic operators, Topol. Methods Nonlinear Anal., 44 (2014), 527-538.  doi: 10.12775/TMNA.2014.059.

[12]

A. Fiscella, R. Servadei and E. Valdinoci, A resonance problem for non-local elliptic operators, Z. Anal. Anwend., 32 (2013), 411-431.  doi: 10.4171/ZAA/1492.

[13]

A. Fiscella, R. Servadei and E. Valdinoci, Asymptotically linear problems driven by fractional Laplacian operators, Math. Methods Appl. Sci., 38 (2015), 3551-3563.  doi: 10.1002/mma.3438.

[14]

A. Iannizzotto and N. S. Papageorgiou, Existence and multiplicity results for resonant fractional boundary value problems, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 511-532.  doi: 10.3934/dcdss.2018028.

[15]

S. Li, K. Perera and J. Su, Computations of critical groups in elliptic boundary value problems where the asymptotic limits may not exist, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 721-732.  doi: 10.1017/S0308210501000324.

[16]

J. Liu, The Morse index for a saddle point, Syst. Sci. Math. Sci., 2 (1989), 32-39. 

[17]

J. Liu and S. Li, An existence theorem for multiple critical points and its application, Kexue Tongbao, 17 (1984), 1025-1027. 

[18]

J. Liu and J. Su, Remarks on multiple nontrivial solutions for quasi-linear resonant problems, J. Math. Anal. Appl., 258 (2001), 209-222.  doi: 10.1006/jmaa.2000.7374.

[19]

J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer, Berlin, 1989. doi: 10.1007/978-1-4757-2061-7.

[20]

G. Molica Bisci, V. D. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Encyclopedia of Mathematics and its Applications, 162. Cambridge University Press, Cambridge, 2016. With a foreword by Jean Mawhin. doi: 10.1017/CBO9781316282397.

[21]

G. Molica Bisci and R. Servadei, A Brezis-Nirenberg splitting approach for nonlocal fractional equations, Nonlinear Anal., 119 (2015), 341-353.  doi: 10.1016/j.na.2014.10.025.

[22]

D. Mugnai and D. Pagliardini, Existence and multiplicity results for the fractional Laplacian in bounded domains, Adv. Calc. Var., 10 (2017), 111-124.  doi: 10.1515/acv-2015-0032.

[23]

K. Perera and M. Schechter, Solution of nonlinear equations having asymptotic limits at zero and infinity, Calc. Var. Partial Differential Equations, 12 (2001), 359-369.  doi: 10.1007/PL00009917.

[24]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory With Application to Differential Equations, CBMS, Vol. 65 AMS: Providence 1986. doi: 10.1090/cbms/065.

[25]

R. Servadei, A critical fractional Laplace equation in the resonant case, Topol. Methods Nonlinear Anal., 43 (2014), 251-267.  doi: 10.12775/TMNA.2014.015.

[26]

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.

[27]

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.

[28]

R. Servadei and E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension, Commun. Pure Appl. Anal., 12 (2013), 2445-2464.  doi: 10.3934/cpaa.2013.12.2445.

[29]

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.

[30]

R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian,, Trans. Amer. Math. Soc., 367 (2015), 67-102.  doi: 10.1090/S0002-9947-2014-05884-4.

[31]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153.

[32]

J. Su, Semilinear elliptic boundary value problems with double resonance between two consecutive eigenvalues,, Nonlinear Anal., 48 (2002), 881-895.  doi: 10.1016/S0362-546X(00)00221-2.

[33]

J. Su, Multiple results for asymptotically linear elliptic problems at resonance, J. Math. Anal. Appl., 278 (2003), 397-408.  doi: 10.1016/S0022-247X(02)00707-2.

[34]

Z.-Q. Wang, Multiple solutions for indefinite functionals and applications to asymptotically linear problems, Acta. Math. Sinica(N.S.), 5 (1989), 101-113.  doi: 10.1007/BF02107664.

show all references

References:
[1]

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.

[2]

M. Badiale and E. Serra, Semilinear Elliptic Equations for Beginners, Springer, Berlin, 2011. doi: 10.1007/978-0-85729-227-8.

[3]

B. Barrios, E. Colorado, A. de Pablo and U. Sánchez, On some critical problems for the fractional Laplacian operator, J. Differential Equations, 252 (2012), 6133-6162.  doi: 10.1016/j.jde.2012.02.023.

[4]

T. Bartsch and S. Li, Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlinear Anal., 28 (1997), 419-441.  doi: 10.1016/0362-546X(95)00167-T.

[5]

K.-C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser-Bosten, 1993. doi: 10.1007/978-1-4612-0385-8.

[6]

Y. Chen and J. Su, Resonant problems for fractional Laplacian, Commun. Pure Appl. Anal., 16 (2017), 163-187.  doi: 10.3934/cpaa.2017008.

[7]

Y. Chen and J. Su, Multiple solutions for the fractional Laplacian problems with different asymptotic limits near infinity, Appl. Math. Lett., 76 (2018), 60-65.  doi: 10.1016/j.aml.2017.07.012.

[8]

Y. Chen and J. Su, Bounded resonant problems driven by fractional Laplacian, Topol. Methods Nonlinear Anal., to appear.

[9]

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.

[10]

M. M. Fall and V. Felli, Unique continuation property and local asymptotics of solutions to fractional elliptic equations, Comm. Partial Differential Equations, 39 (2014), 354-397.  doi: 10.1080/03605302.2013.825918.

[11]

A. Fiscella, Saddle point solutions for nonlocal elliptic operators, Topol. Methods Nonlinear Anal., 44 (2014), 527-538.  doi: 10.12775/TMNA.2014.059.

[12]

A. Fiscella, R. Servadei and E. Valdinoci, A resonance problem for non-local elliptic operators, Z. Anal. Anwend., 32 (2013), 411-431.  doi: 10.4171/ZAA/1492.

[13]

A. Fiscella, R. Servadei and E. Valdinoci, Asymptotically linear problems driven by fractional Laplacian operators, Math. Methods Appl. Sci., 38 (2015), 3551-3563.  doi: 10.1002/mma.3438.

[14]

A. Iannizzotto and N. S. Papageorgiou, Existence and multiplicity results for resonant fractional boundary value problems, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 511-532.  doi: 10.3934/dcdss.2018028.

[15]

S. Li, K. Perera and J. Su, Computations of critical groups in elliptic boundary value problems where the asymptotic limits may not exist, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 721-732.  doi: 10.1017/S0308210501000324.

[16]

J. Liu, The Morse index for a saddle point, Syst. Sci. Math. Sci., 2 (1989), 32-39. 

[17]

J. Liu and S. Li, An existence theorem for multiple critical points and its application, Kexue Tongbao, 17 (1984), 1025-1027. 

[18]

J. Liu and J. Su, Remarks on multiple nontrivial solutions for quasi-linear resonant problems, J. Math. Anal. Appl., 258 (2001), 209-222.  doi: 10.1006/jmaa.2000.7374.

[19]

J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer, Berlin, 1989. doi: 10.1007/978-1-4757-2061-7.

[20]

G. Molica Bisci, V. D. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Encyclopedia of Mathematics and its Applications, 162. Cambridge University Press, Cambridge, 2016. With a foreword by Jean Mawhin. doi: 10.1017/CBO9781316282397.

[21]

G. Molica Bisci and R. Servadei, A Brezis-Nirenberg splitting approach for nonlocal fractional equations, Nonlinear Anal., 119 (2015), 341-353.  doi: 10.1016/j.na.2014.10.025.

[22]

D. Mugnai and D. Pagliardini, Existence and multiplicity results for the fractional Laplacian in bounded domains, Adv. Calc. Var., 10 (2017), 111-124.  doi: 10.1515/acv-2015-0032.

[23]

K. Perera and M. Schechter, Solution of nonlinear equations having asymptotic limits at zero and infinity, Calc. Var. Partial Differential Equations, 12 (2001), 359-369.  doi: 10.1007/PL00009917.

[24]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory With Application to Differential Equations, CBMS, Vol. 65 AMS: Providence 1986. doi: 10.1090/cbms/065.

[25]

R. Servadei, A critical fractional Laplace equation in the resonant case, Topol. Methods Nonlinear Anal., 43 (2014), 251-267.  doi: 10.12775/TMNA.2014.015.

[26]

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.

[27]

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.

[28]

R. Servadei and E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension, Commun. Pure Appl. Anal., 12 (2013), 2445-2464.  doi: 10.3934/cpaa.2013.12.2445.

[29]

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.

[30]

R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian,, Trans. Amer. Math. Soc., 367 (2015), 67-102.  doi: 10.1090/S0002-9947-2014-05884-4.

[31]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153.

[32]

J. Su, Semilinear elliptic boundary value problems with double resonance between two consecutive eigenvalues,, Nonlinear Anal., 48 (2002), 881-895.  doi: 10.1016/S0362-546X(00)00221-2.

[33]

J. Su, Multiple results for asymptotically linear elliptic problems at resonance, J. Math. Anal. Appl., 278 (2003), 397-408.  doi: 10.1016/S0022-247X(02)00707-2.

[34]

Z.-Q. Wang, Multiple solutions for indefinite functionals and applications to asymptotically linear problems, Acta. Math. Sinica(N.S.), 5 (1989), 101-113.  doi: 10.1007/BF02107664.

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