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Traveling wave solutions of a diffusive ratio-dependent Holling-Tanner system with distributed delay
Multiplicity and concentration of solutions for nonlinear fractional elliptic equations with steep potential
Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China |
$\left\{\begin{array}{*{20}{l}}(-Δ)^α u+V_λ(x)u = a(x)|u|^{q-2}u+b(x)|u|^{p-2}u &{\rm in}\,\,\mathbb{R}^N,\\u≥0\,\,&{\rm in}\,\,\mathbb{R}^N, \end{array} \right.$ |
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K. Brown and Y. Zhang,
The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function], J. Differential Equations, 193 (2003), 481-499.
doi: 10.1016/S0022-0396(03)00121-9. |
[2] |
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. |
[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 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. |
[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] |
E. Colorado, A. de Pablo and U. Sánchez,
Perturbations of a critical fractional equation, Pacific J. Math., 271 (2014), 65-85.
|
[7] |
A. Cotsiolis and N. Tavoularis,
Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl., 295 (2004), 225-236.
doi: 10.1016/j.jmaa.2004.03.034. |
[8] |
Y. H. Cheng and T. F. Wu,
Multiplicity and concentration of positive solutions for semilinear elliptic equations with steep potential, Comm. Pure and Applied Ana., 15 (2016), 2457-2473.
doi: 10.3934/cpaa.2016044. |
[9] |
J. Dávila, M. del Pino and J. Wei,
Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differential Equations, 256 (2014), 858-892.
doi: 10.1016/j.jde.2013.10.006. |
[10] |
J. Dávila, M. del Pino, S. Dipierro and E. Valdinoci,
Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum, Anal. PDE, 8 (2015), 1165-1235.
doi: 10.2140/apde.2015.8.1165. |
[11] |
S. Dipierro, M. Medina and E. Valdinoci, Fractional elliptic problems with critical growth in the whole of $\mathbb{R}^N$, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], 15. Edizioni della Normale, Pisa, 2017. ⅷ+152 pp. |
[12] |
P. Drábek and S. Pohozaev,
Positive solutions for the p-Laplacian: application of the fibrering method, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 703-726.
doi: 10.1017/S0308210500023787. |
[13] |
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. |
[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.
doi: 10.1016/0022-247X(74)90025-0. |
[16] |
M. M. Fall, F. Mahmoudi and E. Valdinoci,
Ground states and concentration phenomena for the fractional Schrödinger equation, Nonlinearity, 28 (2015), 1937-1961.
doi: 10.1088/0951-7715/28/6/1937. |
[17] |
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. |
[18] |
N. Laskin,
Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.
doi: 10.1016/S0375-9601(00)00201-2. |
[19] |
N. Laskin,
Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108.
doi: 10.1103/PhysRevE.66.056108. |
[20] |
Z. Nehari,
On a class of nonlinear second-order differential equations, Trans. Amer. Math. Soc., 95 (1960), 101-123.
doi: 10.2307/1993333. |
[21] |
S. I. Pohozaev,
An approach to nonlinear equations (Russian), Dokl. Akad. Nauk SSSR, 247 (1979), 1327-1331.
|
[22] |
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), 40.
doi: 10.1007/s00033-016-0631-5. |
[23] |
P. Rabinowitz, Variational methods for nonlinear eigenvalue problems of nonlinear problems, Edizioni Cremonese, Rome, 1974,139-195. |
[24] |
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. |
[25] |
R. Servadei and E. Valdinoci,
Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.
|
[26] |
G. Tarantello,
On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992), 243-261.
doi: 10.1016/S0294-1449(16)30238-4. |
[27] |
C. Torres,
Existence and concentration of solutions for a non-linear fractional Schrödinger equation with steep potential well, Comm. Pure and Applied Ana., 15 (2016), 535-547.
doi: 10.3934/cpaa.2016.15.535. |
[28] |
T. F. Wu,
Multiple positive solutions for a class of concave-convex elliptic problems in involving sign-changing, J. Funct. Anal., 258 (2010), 99-131.
doi: 10.1016/j.jfa.2009.08.005. |
[29] |
T. F. Wu,
Multiplicity results for a semi-linear elliptic equation involving sign-changing weight function, Rocky Mountain J. Math., 39 (2009), 995-1011.
doi: 10.1216/RMJ-2009-39-3-995. |
[30] |
X. Yu,
The Nehari manifold for elliptic equation involving the square root of the laplacian, J. Differential Equations, 252 (2012), 1283-1308.
doi: 10.1016/j.jde.2011.09.015. |
show all references
References:
[1] |
K. Brown and Y. Zhang,
The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function], J. Differential Equations, 193 (2003), 481-499.
doi: 10.1016/S0022-0396(03)00121-9. |
[2] |
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. |
[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 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. |
[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] |
E. Colorado, A. de Pablo and U. Sánchez,
Perturbations of a critical fractional equation, Pacific J. Math., 271 (2014), 65-85.
|
[7] |
A. Cotsiolis and N. Tavoularis,
Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl., 295 (2004), 225-236.
doi: 10.1016/j.jmaa.2004.03.034. |
[8] |
Y. H. Cheng and T. F. Wu,
Multiplicity and concentration of positive solutions for semilinear elliptic equations with steep potential, Comm. Pure and Applied Ana., 15 (2016), 2457-2473.
doi: 10.3934/cpaa.2016044. |
[9] |
J. Dávila, M. del Pino and J. Wei,
Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differential Equations, 256 (2014), 858-892.
doi: 10.1016/j.jde.2013.10.006. |
[10] |
J. Dávila, M. del Pino, S. Dipierro and E. Valdinoci,
Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum, Anal. PDE, 8 (2015), 1165-1235.
doi: 10.2140/apde.2015.8.1165. |
[11] |
S. Dipierro, M. Medina and E. Valdinoci, Fractional elliptic problems with critical growth in the whole of $\mathbb{R}^N$, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], 15. Edizioni della Normale, Pisa, 2017. ⅷ+152 pp. |
[12] |
P. Drábek and S. Pohozaev,
Positive solutions for the p-Laplacian: application of the fibrering method, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 703-726.
doi: 10.1017/S0308210500023787. |
[13] |
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. |
[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.
doi: 10.1016/0022-247X(74)90025-0. |
[16] |
M. M. Fall, F. Mahmoudi and E. Valdinoci,
Ground states and concentration phenomena for the fractional Schrödinger equation, Nonlinearity, 28 (2015), 1937-1961.
doi: 10.1088/0951-7715/28/6/1937. |
[17] |
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. |
[18] |
N. Laskin,
Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.
doi: 10.1016/S0375-9601(00)00201-2. |
[19] |
N. Laskin,
Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108.
doi: 10.1103/PhysRevE.66.056108. |
[20] |
Z. Nehari,
On a class of nonlinear second-order differential equations, Trans. Amer. Math. Soc., 95 (1960), 101-123.
doi: 10.2307/1993333. |
[21] |
S. I. Pohozaev,
An approach to nonlinear equations (Russian), Dokl. Akad. Nauk SSSR, 247 (1979), 1327-1331.
|
[22] |
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), 40.
doi: 10.1007/s00033-016-0631-5. |
[23] |
P. Rabinowitz, Variational methods for nonlinear eigenvalue problems of nonlinear problems, Edizioni Cremonese, Rome, 1974,139-195. |
[24] |
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. |
[25] |
R. Servadei and E. Valdinoci,
Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.
|
[26] |
G. Tarantello,
On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992), 243-261.
doi: 10.1016/S0294-1449(16)30238-4. |
[27] |
C. Torres,
Existence and concentration of solutions for a non-linear fractional Schrödinger equation with steep potential well, Comm. Pure and Applied Ana., 15 (2016), 535-547.
doi: 10.3934/cpaa.2016.15.535. |
[28] |
T. F. Wu,
Multiple positive solutions for a class of concave-convex elliptic problems in involving sign-changing, J. Funct. Anal., 258 (2010), 99-131.
doi: 10.1016/j.jfa.2009.08.005. |
[29] |
T. F. Wu,
Multiplicity results for a semi-linear elliptic equation involving sign-changing weight function, Rocky Mountain J. Math., 39 (2009), 995-1011.
doi: 10.1216/RMJ-2009-39-3-995. |
[30] |
X. Yu,
The Nehari manifold for elliptic equation involving the square root of the laplacian, J. Differential Equations, 252 (2012), 1283-1308.
doi: 10.1016/j.jde.2011.09.015. |
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