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doi: 10.3934/dcdsb.2020325

On the semilinear fractional elliptic equations with singular weight functions

Ying-Chieh Lin and  Tsung-Fang Wu ,

Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan

* Corresponding author: Tsung-Fang Wu

Received  August 2020 Revised  September 2020 Published  November 2020

Fund Project: Y.J. Lin was supported in part by the Ministry of Science and Technology, Taiwan (Grant No. 109-2115-M-390-001-MY2). T.F. Wu was supported in part by the Ministry of Science and Technology, Taiwan (Grant No. 108-2115-M-390-007-MY2)

In the paper, we study a class of semilinear fractional semilinear elliptic equations involving concave-convex nonlinearities:
$ \begin{equation*} \left\{ \begin{array}{ll} (-\Delta)^{\alpha} u+V_{\lambda }\left( x\right) u = f\left( x\right) \left\vert u\right\vert ^{q-2}u+g\left( x\right) \left\vert u\right\vert ^{p-2}u & \text{in }\mathbb{R}^{N}, \\ u\in H^{\alpha}(\mathbb{R}^{N}), & \end{array}\right. \end{equation*} $
where
$ \alpha\in (0,1] $
,
$ 1<q<2<p<2_{\alpha}^{\ast }\ \left( 2_{\alpha}^{\ast } = \frac{2N}{N-2\alpha}\text{ for}\ N> 2\alpha\right), $
 the potential
$ V_{\lambda }(x) = \lambda a(x)-b(x) $
 and the parameter
$ \lambda >0. $
 Under some suitable assumptions on
$ a,b $
 and the weight functions
$ f,g $
, we obtain the existence and multiplicity of non-trivial (positive) solutions for
$ \lambda $
 large enough. An interesting phenomenon is that we do not need the condition that weight functions
$ f, g $
 are integrable or bounded on whole space
$ \mathbb{R}^{N}. $
Keywords: Semilinear fractional elliptic equations, concave-convex nonlinearities, variational methods, steep potential, singular weight functions.
Mathematics Subject Classification: Primary:35J20;Secondary:35J61.
Citation: Ying-Chieh Lin, Tsung-Fang Wu. On the semilinear fractional elliptic equations with singular weight functions. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020325
References:
[1]

P. Adimurthy, F. Pacella and S. L. Yadava, On the number of positive solutions of some semilinear Dirichlet problems in a ball, Diff. Int. Equations, 10 (1997), 1157-1170.   Google Scholar

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C. O. Alves and G. M. Figueiredo, Multi-bump solutions for a Kirchhoff-type problem, Adv. Nonlinear Anal., 5 (2016), 1-26.  doi: 10.1515/anona-2015-0101.  Google Scholar

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A. Ambrosetti, J. G. Azorero and I. Peral, Elliptic variational problems in $\mathbb{R^N}$ with critical growth, J. Differential Equations, 168 (2000), 10-32.  doi: 10.1006/jdeq.2000.3875.  Google Scholar

[5]

A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.  doi: 10.1006/jfan.1994.1078.  Google Scholar

[6]

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

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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.  Google Scholar

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T. Bartsch, A. Pankov and Z.-Q. Wang, Nonlinear Schrödinger equations with steep potential well, Commun. Contemp. Math., 3 (2001), 549-569.  doi: 10.1142/S0219199701000494.  Google Scholar

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T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbb{R}^{N}$, Comm. Partial Differential Equations, 20 (1995), 1725-1741.  doi: 10.1080/03605309508821149.  Google Scholar

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P. A. Binding, P. Drábek and Y. X. Huang, On Neumann boundary value problems for some quasilinear elliptic equations, Electr. J. Diff. Eqns., (1997), 11 pp.  Google Scholar

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K. J. Brown andd T.-F. Wu, A fibering map approach to a semilinear elliptic boundary value problem, Electr. J. Diff. Eqns., (2007), 9 pp.  Google Scholar

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K. J. Brown and T. F. Wu, A fibering map approach to a potential operator equation and its applications, Diff. Int. Equations, 22 (2009), 1097-1114.   Google Scholar

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K. J. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Diff. Equns, 193 (2003), 481-499.  doi: 10.1016/S0022-0396(03)00121-9.  Google Scholar

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L. Caffarelli, S. Dipierro and E. Valdinoci, A logistic equation with nonlocal interactions, Kinet. Relat. Models, 10 (2017), 141-170.  doi: 10.3934/krm.2017006.  Google Scholar

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J. Chabrowski and João Marcos Bezzera do Ó, On semilinear elliptic equations involving concave and convex nonlinearities, Math. Nachr., 233/234 (2002), 55-76.  doi: 10.1002/1522-2616(200201)233:1<55::AID-MANA55>3.0.CO;2-R.  Google Scholar

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Y.-H. Cheng and T. F. Wu, Multiplicity and concentration of positive solutions for semilinear elliptic equaitons with steep potential, Commun. Pure Appl. Anal., 15 (2016), 2457-2473.  doi: 10.3934/cpaa.2016044.  Google Scholar

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L. Damascelli, M. Grossi and F. Pacella, Qualitative properties of positive solutions of semilinear elliptic equations in symmetric domains via the maximum principle, Annls Inst. H. Poincaré Analyse Non linéaire, 16 (1999), 631-652.  doi: 10.1016/S0294-1449(99)80030-4.  Google Scholar

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E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhikers guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[22]

P. Drábek and S. I. Pohozaev, Positive solutions for the $p$-Laplacian: Application of the fibering method, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 703-726.  doi: 10.1017/S0308210500023787.  Google Scholar

[23]

I. Ekeland, On the variational principle, J. Math. Anal. Appl., 17 (1974), 324-353.  doi: 10.1016/0022-247X(74)90025-0.  Google Scholar

[24]

A. Elgart and B. Schlein, Mean field dynamics of boson stars, Comm. Pure Appl. Math., 60 (2007), 500-545.  doi: 10.1002/cpa.20134.  Google Scholar

[25]

P. Felmer, A. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A., 142 (2012), 1237-1262.  doi: 10.1017/S0308210511000746.  Google Scholar

[26]

J. Fhlich, B. L. G. Jonsson and E. Lenzmann, Boson stars as solitary waves, Comm. Math. Phys., 274 (2007), 1-30.  doi: 10.1007/s00220-007-0272-9.  Google Scholar

[27]

D. G. de Figueiredo, J. P. Gossez and P. Ubilla, Local superlinearity and sublinearity for indefinite semilinear elliptic problems, J. Funct. Anal., 199 (2003), 452-467.  doi: 10.1016/S0022-1236(02)00060-5.  Google Scholar

[28]

J. Frhlich and E. Lenzmann, Blowup for nonlinear wave equations describing boson stars, Comm. Pure Appl. Math., 60 (2007), 1691-1705.  doi: 10.1002/cpa.20186.  Google Scholar

[29]

J. V. Goncalves and O. H. Miyagaki, Multiple positive solutions for semilinear elliptic equations in $\mathbb{R}^{N}$ involving subcritical exponents, Nonlinear Analysis, 32 (1998), 41-51.  doi: 10.1016/S0362-546X(97)00451-3.  Google Scholar

[30]

T.-S. Hsu and H. L. Lin, Multiple positive solutions for semilinear elliptic equations in $\mathbb{R}^{N}$ involving concave-convex nonlineatlties and sign-changing weight functions, Abstract and Applied Analysis, 2010 (2010), Art. ID 658397, 21 pp. doi: 10.1155/2010/658397.  Google Scholar

[31]

N. Laskin, Fractional quantum mechanics and Lvy path integral, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.  Google Scholar

[32]

N. Laskin, Fractional Schrödinger equation, Phys. Rev. E., 66 (2002), 056108, 7 pp. doi: 10.1103/PhysRevE.66.056108.  Google Scholar

[33]

F.-F. Liao and C.-L. Tang, Four positive solutions of a quasilinear elliptic equation in $\mathbb{R}^{N},$, Comm. Pure Appl. Anal., 12 (2013), 2577-2600.  doi: 10.3934/cpaa.2013.12.2577.  Google Scholar

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E. H. Lieb and H.-T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Comm. Math. Phys., 112 (1987), 147-174.  doi: 10.1007/BF01217684.  Google Scholar

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P.-L. Lions, The concentration-compactness principle in the calculus of variations. The local compact case. I, Ann. Inst. H. Poincar é Anal. Non Lineairé, 1 (1984), 109-145.  doi: 10.1016/S0294-1449(16)30428-0.  Google Scholar

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Z. Liu and Z.-Q. Wang, Schrödinger equations with concave and convex nonlinearities, Z. Angew. Math. Phys., 56 (2005), 609-629.  doi: 10.1007/s00033-005-3115-6.  Google Scholar

[37]

S. Mao and A. Xia, Multiplicity results of nonlinear fractional magnetic Schrödinger equation with steep potential, Appl. Math. Lett., 97 (2019), 73-80.  doi: 10.1016/j.aml.2019.05.027.  Google Scholar

[38]

A. Massaccesi and E. Valdinoci, Is a nonlocal diffusion strategy convenient for biological populations in competition?, J. Math. Biol., 74 (2017), 113-147.  doi: 10.1007/s00285-016-1019-z.  Google Scholar

[39]

T. Ouyang and J. Shi, Exact multiplicity of positive solutions for a class of semilinear problem Ⅱ, J. Diff. Eqns., 158 (1999), 94-151.  doi: 10.1016/S0022-0396(99)80020-5.  Google Scholar

[40]

F. O. de Paiva, Nonnegative solutions of elliptic problems with sublinear indefinite nonlinearity, J. Func. Anal., 261 (2011), 2569-2586.  doi: 10.1016/j.jfa.2011.07.002.  Google Scholar

[41]

S. Peng and A. Xia, Multiplicity and concentration of solutions for nonlinear fractional elliptic equations with steep potential, Commun. Pure Appl. Anal., 17 (2018), 1201-1217.  doi: 10.3934/cpaa.2018058.  Google Scholar

[42]

A. Quaas and A. Xia, Multiple positive solutions for nonlinear critical fractional elliptic equations involving sign-changing weight functions, Z. Angew. Math. Phys., 67 (2016), Art. 40, 21 pp. doi: 10.1007/s00033-016-0631-5.  Google Scholar

[43]

J. Sun and T.-F. Wu, Ground state solutions for an indefinite Kirchhoff type problem with steep potential well, J. Differential Equations, 256 (2014), 1771-1792.  doi: 10.1016/j.jde.2013.12.006.  Google Scholar

[44]

M. Tang, Exact multiplicity for semilinear elliptic Dirichlet problems involving concave and convex nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 705-717.  doi: 10.1017/S0308210500002614.  Google Scholar

[45]

Q. Wang, The Nehari manifold for a fractional Laplacian equation involving critical nonlinearities, Commun. Pure Appl. Anal., 17 (2018), 2261-2281.  doi: 10.3934/cpaa.2018108.  Google Scholar

[46]

T. F. Wu, On semilinear elliptic equations involving concave–convex nonlinearities and sign-changing weight function, J. Math. Anal. Appl., 318 (2006), 253-270.  doi: 10.1016/j.jmaa.2005.05.057.  Google Scholar

[47]

T.-F. Wu, Multiplicity of positive solutions for semilinear elliptic equations in $\mathbb{R}^{N}$, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 647-670.  doi: 10.1017/S0308210506001156.  Google Scholar

[48]

T.-F. Wu, Three positive solutions for Dirichlet problems involving critical Sobolev exponent and sign-changing weight, J. Differ. Equat., 249 (2010), 1459-1578.  doi: 10.1016/j.jde.2010.07.021.  Google Scholar

[49]

T. F. Wu, Multiple positive solutions for a class of concave-convex elliptic problem in $\mathbb{R}^{N}$ involving sign-changing weight, J. Funct. Anal., 258 (2010), 99-131.  doi: 10.1016/j.jfa.2009.08.005.  Google Scholar

[50]

H. Yin, Z. Yang and Z. Feng, Multiple positive solutions for a quasilinear elliptic equation in $\mathbb{R}^{N}$, Diff. Integ. Eqns, 25 (2012), 977-992.   Google Scholar

[51]

L. Zhao, H. Liu and F. Zhao, Existence and concentration of solutions for the Schrödinger–Poisson equations with steep well potential, J. Diff. Eqns., 255 (2013), 1-23.  doi: 10.1016/j.jde.2013.03.005.  Google Scholar

show all references

References:
[1]

P. Adimurthy, F. Pacella and S. L. Yadava, On the number of positive solutions of some semilinear Dirichlet problems in a ball, Diff. Int. Equations, 10 (1997), 1157-1170.   Google Scholar

[2]

C. O. Alves and G. M. Figueiredo, Multi-bump solutions for a Kirchhoff-type problem, Adv. Nonlinear Anal., 5 (2016), 1-26.  doi: 10.1515/anona-2015-0101.  Google Scholar

[3]

A. Ambrosetti, J. G. Azorero and I. Peral, Multiplicity results for some nonlinear elliptic equations, J. Funct. Anal., 137 (1996), 219-242.  doi: 10.1006/jfan.1996.0045.  Google Scholar

[4]

A. Ambrosetti, J. G. Azorero and I. Peral, Elliptic variational problems in $\mathbb{R^N}$ with critical growth, J. Differential Equations, 168 (2000), 10-32.  doi: 10.1006/jdeq.2000.3875.  Google Scholar

[5]

A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.  doi: 10.1006/jfan.1994.1078.  Google Scholar

[6]

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

[7]

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.  Google Scholar

[8]

T. Bartsch, A. Pankov and Z.-Q. Wang, Nonlinear Schrödinger equations with steep potential well, Commun. Contemp. Math., 3 (2001), 549-569.  doi: 10.1142/S0219199701000494.  Google Scholar

[9]

T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbb{R}^{N}$, Comm. Partial Differential Equations, 20 (1995), 1725-1741.  doi: 10.1080/03605309508821149.  Google Scholar

[10]

H. Berestycki, J.-M. Roquejoffre and L. Rossi, The periodic patch model for population dynamics with fractional diffusion, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 1-13.  doi: 10.3934/dcdss.2011.4.1.  Google Scholar

[11]

P. A. Binding, P. Drábek and Y. X. Huang, On Neumann boundary value problems for some quasilinear elliptic equations, Electr. J. Diff. Eqns., (1997), 11 pp.  Google Scholar

[12]

K. J. Brown andd T.-F. Wu, A fibering map approach to a semilinear elliptic boundary value problem, Electr. J. Diff. Eqns., (2007), 9 pp.  Google Scholar

[13]

K. J. Brown and T. F. Wu, A fibering map approach to a potential operator equation and its applications, Diff. Int. Equations, 22 (2009), 1097-1114.   Google Scholar

[14]

K. J. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Diff. Equns, 193 (2003), 481-499.  doi: 10.1016/S0022-0396(03)00121-9.  Google Scholar

[15]

L. Caffarelli, S. Dipierro and E. Valdinoci, A logistic equation with nonlocal interactions, Kinet. Relat. Models, 10 (2017), 141-170.  doi: 10.3934/krm.2017006.  Google Scholar

[16]

J. Chabrowski and João Marcos Bezzera do Ó, On semilinear elliptic equations involving concave and convex nonlinearities, Math. Nachr., 233/234 (2002), 55-76.  doi: 10.1002/1522-2616(200201)233:1<55::AID-MANA55>3.0.CO;2-R.  Google Scholar

[17]

C.-Y. Chen and T.-F. Wu, Multiple positive solutions for indefinite semilinear elliptic problems involving a critical Sobolev exponent, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 691-709.  doi: 10.1017/S0308210512000133.  Google Scholar

[18]

Y.-H. Cheng and T. F. Wu, Multiplicity and concentration of positive solutions for semilinear elliptic equaitons with steep potential, Commun. Pure Appl. Anal., 15 (2016), 2457-2473.  doi: 10.3934/cpaa.2016044.  Google Scholar

[19]

R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC Financ. Math. Ser., Chapman & Hall/CRC, Boca Raton, FL, 2004.  Google Scholar

[20]

L. Damascelli, M. Grossi and F. Pacella, Qualitative properties of positive solutions of semilinear elliptic equations in symmetric domains via the maximum principle, Annls Inst. H. Poincaré Analyse Non linéaire, 16 (1999), 631-652.  doi: 10.1016/S0294-1449(99)80030-4.  Google Scholar

[21]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhikers guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[22]

P. Drábek and S. I. Pohozaev, Positive solutions for the $p$-Laplacian: Application of the fibering method, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 703-726.  doi: 10.1017/S0308210500023787.  Google Scholar

[23]

I. Ekeland, On the variational principle, J. Math. Anal. Appl., 17 (1974), 324-353.  doi: 10.1016/0022-247X(74)90025-0.  Google Scholar

[24]

A. Elgart and B. Schlein, Mean field dynamics of boson stars, Comm. Pure Appl. Math., 60 (2007), 500-545.  doi: 10.1002/cpa.20134.  Google Scholar

[25]

P. Felmer, A. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A., 142 (2012), 1237-1262.  doi: 10.1017/S0308210511000746.  Google Scholar

[26]

J. Fhlich, B. L. G. Jonsson and E. Lenzmann, Boson stars as solitary waves, Comm. Math. Phys., 274 (2007), 1-30.  doi: 10.1007/s00220-007-0272-9.  Google Scholar

[27]

D. G. de Figueiredo, J. P. Gossez and P. Ubilla, Local superlinearity and sublinearity for indefinite semilinear elliptic problems, J. Funct. Anal., 199 (2003), 452-467.  doi: 10.1016/S0022-1236(02)00060-5.  Google Scholar

[28]

J. Frhlich and E. Lenzmann, Blowup for nonlinear wave equations describing boson stars, Comm. Pure Appl. Math., 60 (2007), 1691-1705.  doi: 10.1002/cpa.20186.  Google Scholar

[29]

J. V. Goncalves and O. H. Miyagaki, Multiple positive solutions for semilinear elliptic equations in $\mathbb{R}^{N}$ involving subcritical exponents, Nonlinear Analysis, 32 (1998), 41-51.  doi: 10.1016/S0362-546X(97)00451-3.  Google Scholar

[30]

T.-S. Hsu and H. L. Lin, Multiple positive solutions for semilinear elliptic equations in $\mathbb{R}^{N}$ involving concave-convex nonlineatlties and sign-changing weight functions, Abstract and Applied Analysis, 2010 (2010), Art. ID 658397, 21 pp. doi: 10.1155/2010/658397.  Google Scholar

[31]

N. Laskin, Fractional quantum mechanics and Lvy path integral, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.  Google Scholar

[32]

N. Laskin, Fractional Schrödinger equation, Phys. Rev. E., 66 (2002), 056108, 7 pp. doi: 10.1103/PhysRevE.66.056108.  Google Scholar

[33]

F.-F. Liao and C.-L. Tang, Four positive solutions of a quasilinear elliptic equation in $\mathbb{R}^{N},$, Comm. Pure Appl. Anal., 12 (2013), 2577-2600.  doi: 10.3934/cpaa.2013.12.2577.  Google Scholar

[34]

E. H. Lieb and H.-T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Comm. Math. Phys., 112 (1987), 147-174.  doi: 10.1007/BF01217684.  Google Scholar

[35]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The local compact case. I, Ann. Inst. H. Poincar é Anal. Non Lineairé, 1 (1984), 109-145.  doi: 10.1016/S0294-1449(16)30428-0.  Google Scholar

[36]

Z. Liu and Z.-Q. Wang, Schrödinger equations with concave and convex nonlinearities, Z. Angew. Math. Phys., 56 (2005), 609-629.  doi: 10.1007/s00033-005-3115-6.  Google Scholar

[37]

S. Mao and A. Xia, Multiplicity results of nonlinear fractional magnetic Schrödinger equation with steep potential, Appl. Math. Lett., 97 (2019), 73-80.  doi: 10.1016/j.aml.2019.05.027.  Google Scholar

[38]

A. Massaccesi and E. Valdinoci, Is a nonlocal diffusion strategy convenient for biological populations in competition?, J. Math. Biol., 74 (2017), 113-147.  doi: 10.1007/s00285-016-1019-z.  Google Scholar

[39]

T. Ouyang and J. Shi, Exact multiplicity of positive solutions for a class of semilinear problem Ⅱ, J. Diff. Eqns., 158 (1999), 94-151.  doi: 10.1016/S0022-0396(99)80020-5.  Google Scholar

[40]

F. O. de Paiva, Nonnegative solutions of elliptic problems with sublinear indefinite nonlinearity, J. Func. Anal., 261 (2011), 2569-2586.  doi: 10.1016/j.jfa.2011.07.002.  Google Scholar

[41]

S. Peng and A. Xia, Multiplicity and concentration of solutions for nonlinear fractional elliptic equations with steep potential, Commun. Pure Appl. Anal., 17 (2018), 1201-1217.  doi: 10.3934/cpaa.2018058.  Google Scholar

[42]

A. Quaas and A. Xia, Multiple positive solutions for nonlinear critical fractional elliptic equations involving sign-changing weight functions, Z. Angew. Math. Phys., 67 (2016), Art. 40, 21 pp. doi: 10.1007/s00033-016-0631-5.  Google Scholar

[43]

J. Sun and T.-F. Wu, Ground state solutions for an indefinite Kirchhoff type problem with steep potential well, J. Differential Equations, 256 (2014), 1771-1792.  doi: 10.1016/j.jde.2013.12.006.  Google Scholar

[44]

M. Tang, Exact multiplicity for semilinear elliptic Dirichlet problems involving concave and convex nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 705-717.  doi: 10.1017/S0308210500002614.  Google Scholar

[45]

Q. Wang, The Nehari manifold for a fractional Laplacian equation involving critical nonlinearities, Commun. Pure Appl. Anal., 17 (2018), 2261-2281.  doi: 10.3934/cpaa.2018108.  Google Scholar

[46]

T. F. Wu, On semilinear elliptic equations involving concave–convex nonlinearities and sign-changing weight function, J. Math. Anal. Appl., 318 (2006), 253-270.  doi: 10.1016/j.jmaa.2005.05.057.  Google Scholar

[47]

T.-F. Wu, Multiplicity of positive solutions for semilinear elliptic equations in $\mathbb{R}^{N}$, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 647-670.  doi: 10.1017/S0308210506001156.  Google Scholar

[48]

T.-F. Wu, Three positive solutions for Dirichlet problems involving critical Sobolev exponent and sign-changing weight, J. Differ. Equat., 249 (2010), 1459-1578.  doi: 10.1016/j.jde.2010.07.021.  Google Scholar

[49]

T. F. Wu, Multiple positive solutions for a class of concave-convex elliptic problem in $\mathbb{R}^{N}$ involving sign-changing weight, J. Funct. Anal., 258 (2010), 99-131.  doi: 10.1016/j.jfa.2009.08.005.  Google Scholar

[50]

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