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Multiplicity and concentration of positive solutions for semilinear elliptic equations with steep potential
1. | Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan |
References:
[1] |
A. Ambrosetti, G. J. Azorero and I. Peral, Multiplicity results for some nonlinear elliptic equations, J. Funct. Anal., 137 (1996), 219-242.
doi: 10.1006/jfan.1996.0045. |
[2] |
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. |
[3] |
Adimurthy, F. Pacella and L. Yadava, On the number of positive solutions of some semilinear Dirichlet problems in a ball, Diff. Int. Equations, 10 (1997), 1157-1170. |
[4] |
P. A. Binding, P. Drábek and Y. X. Huang, On Neumann boundary value problems for some quasilinear elliptic equations, Electr. J. Diff. Eqns., 5 (1997), 1-11. |
[5] |
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. |
[6] |
T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbbR^N$, Comm. Partial Differential Equations, 20 (1995), 1725-1741.
doi: 10.1080/03605309508821149. |
[7] |
K. J. Brown and T. F. Wu, A fibering map approach to a semilinear elliptic boundary value problem, Electr. J. Diff. Eqns., 69 (2007), 1-9. |
[8] |
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. |
[9] |
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. |
[10] |
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>3.3.CO;2-I. |
[11] |
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., 144A (2014), 691-709.
doi: 10.1017/S0308210512000133. |
[12] |
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. |
[13] |
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. |
[14] |
Y. H. Ding and A. Szulkin, Bound states for semilinear Schrödinger equations with sign-changing potential, Calc. Var. Partial Differential Equations, 29 (2007), 397-419.
doi: 10.1007/s00526-006-0071-8. |
[15] |
I. Ekeland, On the variational principle, J. Math. Anal. Appl., 17 (1974), 324-353. |
[16] |
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. |
[17] |
J. V. Goncalves and O. H. Miyagaki, Multiple positive solutions for semilinear elliptic equations in $\mathbbR^N$ involving subcritical exponents, Nonlinear Analysis: T. M. A., 32 (1998), 41-51.
doi: 10.1016/S0362-546X(97)00451-3. |
[18] |
T. S. Hsu and H. L. Lin, Multiple positive solutions for semilinear elliptic equations in $\mathbbR^N$ involving concave-convex nonlineatlties and sign-changing weight functions, Abstract and Applied Analysis, 2010 ID658397 (2010), 21 pages. |
[19] |
P. L. Lions, The concentration-compactness principle in the calculus of variations. The local compact case Part I, Ann. Inst. H. Poincaré Anal. Non Lineairé, 1 (1984), 109-145. |
[20] |
F. F. Liao and C. L. Tang, Four positive solutions of a quasilinear elliptic equation in $\mathbbR^N$, Comm. Pure Appl. Anal., 12 (2013), 2577-2600.
doi: 10.3934/cpaa.2013.12.2577. |
[21] |
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. |
[22] |
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. |
[23] |
Francisco Odair 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. |
[24] |
M. Struwe, Variational Methods, 2nd edition, Springer-Verlag, Berlin-Heidelberg, 1996.
doi: 10.1007/978-3-662-03212-1. |
[25] |
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. |
[26] |
T. F. Wu, Multiplicity of positive solutions for semilinear elliptic equations in $\mathbbR^N$, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 647-670.
doi: 10.1017/S0308210506001156. |
[27] |
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. |
[28] |
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. |
[29] |
T. F. Wu, Multiple positive solutions for a class of concave-convex elliptic problem in $\mathbbR^N$ involving sign-changing weight, J. Funct. Anal., 258 (2010), 99-131.
doi: 10.1016/j.jfa.2009.08.005. |
[30] |
H. Yin, Z. Yang and Z. Feng, Multiple positive solutions for a quasilinear elliptic equation in $\mathbbR^N$, Diff. Integ. Eqns, 25 (2012), 977-992. |
[31] |
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] |
A. Ambrosetti, G. J. Azorero and I. Peral, Multiplicity results for some nonlinear elliptic equations, J. Funct. Anal., 137 (1996), 219-242.
doi: 10.1006/jfan.1996.0045. |
[2] |
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. |
[3] |
Adimurthy, F. Pacella and L. Yadava, On the number of positive solutions of some semilinear Dirichlet problems in a ball, Diff. Int. Equations, 10 (1997), 1157-1170. |
[4] |
P. A. Binding, P. Drábek and Y. X. Huang, On Neumann boundary value problems for some quasilinear elliptic equations, Electr. J. Diff. Eqns., 5 (1997), 1-11. |
[5] |
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. |
[6] |
T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbbR^N$, Comm. Partial Differential Equations, 20 (1995), 1725-1741.
doi: 10.1080/03605309508821149. |
[7] |
K. J. Brown and T. F. Wu, A fibering map approach to a semilinear elliptic boundary value problem, Electr. J. Diff. Eqns., 69 (2007), 1-9. |
[8] |
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. |
[9] |
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. |
[10] |
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>3.3.CO;2-I. |
[11] |
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., 144A (2014), 691-709.
doi: 10.1017/S0308210512000133. |
[12] |
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. |
[13] |
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. |
[14] |
Y. H. Ding and A. Szulkin, Bound states for semilinear Schrödinger equations with sign-changing potential, Calc. Var. Partial Differential Equations, 29 (2007), 397-419.
doi: 10.1007/s00526-006-0071-8. |
[15] |
I. Ekeland, On the variational principle, J. Math. Anal. Appl., 17 (1974), 324-353. |
[16] |
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. |
[17] |
J. V. Goncalves and O. H. Miyagaki, Multiple positive solutions for semilinear elliptic equations in $\mathbbR^N$ involving subcritical exponents, Nonlinear Analysis: T. M. A., 32 (1998), 41-51.
doi: 10.1016/S0362-546X(97)00451-3. |
[18] |
T. S. Hsu and H. L. Lin, Multiple positive solutions for semilinear elliptic equations in $\mathbbR^N$ involving concave-convex nonlineatlties and sign-changing weight functions, Abstract and Applied Analysis, 2010 ID658397 (2010), 21 pages. |
[19] |
P. L. Lions, The concentration-compactness principle in the calculus of variations. The local compact case Part I, Ann. Inst. H. Poincaré Anal. Non Lineairé, 1 (1984), 109-145. |
[20] |
F. F. Liao and C. L. Tang, Four positive solutions of a quasilinear elliptic equation in $\mathbbR^N$, Comm. Pure Appl. Anal., 12 (2013), 2577-2600.
doi: 10.3934/cpaa.2013.12.2577. |
[21] |
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. |
[22] |
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. |
[23] |
Francisco Odair 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. |
[24] |
M. Struwe, Variational Methods, 2nd edition, Springer-Verlag, Berlin-Heidelberg, 1996.
doi: 10.1007/978-3-662-03212-1. |
[25] |
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. |
[26] |
T. F. Wu, Multiplicity of positive solutions for semilinear elliptic equations in $\mathbbR^N$, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 647-670.
doi: 10.1017/S0308210506001156. |
[27] |
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. |
[28] |
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. |
[29] |
T. F. Wu, Multiple positive solutions for a class of concave-convex elliptic problem in $\mathbbR^N$ involving sign-changing weight, J. Funct. Anal., 258 (2010), 99-131.
doi: 10.1016/j.jfa.2009.08.005. |
[30] |
H. Yin, Z. Yang and Z. Feng, Multiple positive solutions for a quasilinear elliptic equation in $\mathbbR^N$, Diff. Integ. Eqns, 25 (2012), 977-992. |
[31] |
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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