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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  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 & Continuous Dynamical Systems - S, 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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[6]

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

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[17]

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

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[6]

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

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[17]

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

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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