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On the semilinear fractional elliptic equations with singular weight functions
Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan |
$ \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*} $ |
$ \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), $ |
$ V_{\lambda }(x) = \lambda a(x)-b(x) $ |
$ \lambda >0. $ |
$ a,b $ |
$ f,g $ |
$ \lambda $ |
$ f, g $ |
$ \mathbb{R}^{N}. $ |
References:
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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.
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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. |
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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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
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[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. |
[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. |
[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.
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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.
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[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. |
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R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC Financ. Math. Ser., Chapman & Hall/CRC, Boca Raton, FL, 2004. |
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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. |
[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. |
[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. |
[23] |
I. Ekeland,
On the variational principle, J. Math. Anal. Appl., 17 (1974), 324-353.
doi: 10.1016/0022-247X(74)90025-0. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[32] |
N. Laskin, Fractional Schrödinger equation, Phys. Rev. E., 66 (2002), 056108, 7 pp.
doi: 10.1103/PhysRevE.66.056108. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[19] |
R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC Financ. Math. Ser., Chapman & Hall/CRC, Boca Raton, FL, 2004. |
[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. |
[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. |
[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. |
[23] |
I. Ekeland,
On the variational principle, J. Math. Anal. Appl., 17 (1974), 324-353.
doi: 10.1016/0022-247X(74)90025-0. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[32] |
N. Laskin, Fractional Schrödinger equation, Phys. Rev. E., 66 (2002), 056108, 7 pp.
doi: 10.1103/PhysRevE.66.056108. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
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