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Existence of solutions for a Kirchhoff-type-nonlocal operators of elliptic type
1. | Department of Mathematics, Faculty of Sciences, Razi University, Kermanshah, 67149, Iran |
2. | Department of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi, 030024, China |
References:
[1] |
B. Barrios, E. Colorado, A. De Pablo and U. Sanchez, On some critical problems for the fractional Laplacian operator, J. Differential Equations, 252 (2012), 6133-6162.
doi: 10.1016/j.jde.2012.02.023. |
[2] |
X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093.
doi: 10.1016/j.aim.2010.01.025. |
[3] |
A. Capella, Solutions of a pure critical exponent problem involving the half-Laplacian in annularshaped domains, Commun. Pure Appl. Anal., 10 (2011), 1645-1662.
doi: 10.3934/cpaa.2011.10.1645. |
[4] |
A. Fiscella, R. Servadei and E. Valdinoci, A resonance problem for non-local elliptic operators, preprint, available at http://www.ma.utexas.edu/mparc-bin/mpa?yn=12-61. |
[5] |
A. Fiscella, R. Servadei and E. Valdinoci, Asymptotically linear problems driven by the fractional Laplacian operator, preprint, available at http://www.ma.utexas.edu/mp arc-bin/mpa?yn=12-128. |
[6] |
A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal., 94 (2014), 156-170.
doi: 10.1016/j.na.2013.08.011. |
[7] |
J. Liu, The Morse index of a saddle point, Syst. Sci. Math. Sci., 2 (1989), 32-39. |
[8] |
J. Liu and J. Su, Remarks on multiple nontrivial solutions for quasilinear resonant problems, J. Math. Anal. Appl., 258 (2001), 209-222.
doi: 10.1006/jmaa.2000.7374. |
[9] |
R. Servadei, A critical fractional Laplace equation in the resonant case, Topol. Methods Nonlinear Anal., to appear. |
[10] |
R. Servadei, Infinitely many solutions for fractional Laplace equations with subcritical nonlinearity, Contemp. Math., to appear.
doi: 10.1090/conm/595/11809. |
[11] |
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. |
[12] |
R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Continuous Dynam. Systems, 33 (2013), 2105-2137. |
[13] |
R. Servadei and E. Valdinoci, Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators, Rev. Mat. Iberoam., 29 (2013).
doi: 10.4171/RMI/750. |
[14] |
R. Servadei and E. Valdinoci, The Brézis-Nirenberg result for the fractional Laplacian, Trans. AMS, to appear. |
[15] |
R. Servadei and E. Valdinoci, Fractional Laplacian equations with critical Sobolev exponent, preprint, available at http://www.math.utexas.edu/mp arc-bin/mpa?yn=12-58. |
[16] |
G. Sun and K. Teng, Existence and multiplicity of solutions for a class of fractional Kirchhoff-type problem, Math. Commu., 19 (2014), 183-194. |
[17] |
J. Tan, The Brézis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Partial Differential Equations, 36 (2011), 21-41.
doi: 10.1007/s00526-010-0378-3. |
[18] |
K. Teng, Multiplicity results for hemivariational inequalities driven by nonlocal elliptic operators, J. Math. Anal. Appl., 396 (2012), 386-395.
doi: 10.1016/j.jmaa.2012.06.041. |
[19] |
K. Teng, Two nontrivial solutions for hemivariational inequalities driven by nonlocal elliptic operators, Nonlinear Anal. RWA, 14 (2013), 867-874.
doi: 10.1016/j.nonrwa.2012.08.008. |
show all references
References:
[1] |
B. Barrios, E. Colorado, A. De Pablo and U. Sanchez, On some critical problems for the fractional Laplacian operator, J. Differential Equations, 252 (2012), 6133-6162.
doi: 10.1016/j.jde.2012.02.023. |
[2] |
X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093.
doi: 10.1016/j.aim.2010.01.025. |
[3] |
A. Capella, Solutions of a pure critical exponent problem involving the half-Laplacian in annularshaped domains, Commun. Pure Appl. Anal., 10 (2011), 1645-1662.
doi: 10.3934/cpaa.2011.10.1645. |
[4] |
A. Fiscella, R. Servadei and E. Valdinoci, A resonance problem for non-local elliptic operators, preprint, available at http://www.ma.utexas.edu/mparc-bin/mpa?yn=12-61. |
[5] |
A. Fiscella, R. Servadei and E. Valdinoci, Asymptotically linear problems driven by the fractional Laplacian operator, preprint, available at http://www.ma.utexas.edu/mp arc-bin/mpa?yn=12-128. |
[6] |
A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal., 94 (2014), 156-170.
doi: 10.1016/j.na.2013.08.011. |
[7] |
J. Liu, The Morse index of a saddle point, Syst. Sci. Math. Sci., 2 (1989), 32-39. |
[8] |
J. Liu and J. Su, Remarks on multiple nontrivial solutions for quasilinear resonant problems, J. Math. Anal. Appl., 258 (2001), 209-222.
doi: 10.1006/jmaa.2000.7374. |
[9] |
R. Servadei, A critical fractional Laplace equation in the resonant case, Topol. Methods Nonlinear Anal., to appear. |
[10] |
R. Servadei, Infinitely many solutions for fractional Laplace equations with subcritical nonlinearity, Contemp. Math., to appear.
doi: 10.1090/conm/595/11809. |
[11] |
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. |
[12] |
R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Continuous Dynam. Systems, 33 (2013), 2105-2137. |
[13] |
R. Servadei and E. Valdinoci, Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators, Rev. Mat. Iberoam., 29 (2013).
doi: 10.4171/RMI/750. |
[14] |
R. Servadei and E. Valdinoci, The Brézis-Nirenberg result for the fractional Laplacian, Trans. AMS, to appear. |
[15] |
R. Servadei and E. Valdinoci, Fractional Laplacian equations with critical Sobolev exponent, preprint, available at http://www.math.utexas.edu/mp arc-bin/mpa?yn=12-58. |
[16] |
G. Sun and K. Teng, Existence and multiplicity of solutions for a class of fractional Kirchhoff-type problem, Math. Commu., 19 (2014), 183-194. |
[17] |
J. Tan, The Brézis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Partial Differential Equations, 36 (2011), 21-41.
doi: 10.1007/s00526-010-0378-3. |
[18] |
K. Teng, Multiplicity results for hemivariational inequalities driven by nonlocal elliptic operators, J. Math. Anal. Appl., 396 (2012), 386-395.
doi: 10.1016/j.jmaa.2012.06.041. |
[19] |
K. Teng, Two nontrivial solutions for hemivariational inequalities driven by nonlocal elliptic operators, Nonlinear Anal. RWA, 14 (2013), 867-874.
doi: 10.1016/j.nonrwa.2012.08.008. |
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