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On general fractional abstract Cauchy problem
Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent
1. | School of Mathematics and Statistics, Southwest University, Chongqing 400715, China, China |
References:
[1] |
A. M. Mao and Z. T. Zhang, Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition, Nonlinear Anal., 70 (2009), 1275-1287.
doi: 10.1016/j.na.2008.02.011. |
[2] |
A. Hamydy, M. Massar and N. Tsouli, Existence of solution for p-Kirchhoff type problems with critical exponents, Electronic J. Differential Equations, 105 (2011), 1-8. |
[3] |
A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 41 (1973), 349-381.
doi: 10.1016/0022-1236(73)90051-7. |
[4] |
B. Cheng, New existence and multiplicity of nontrivial solutions for nonlocal elliptic Kirchhoff type problems, J. Math. Anal. Appl., 394 (2012), 488-495.
doi: 10.1016/j.jmaa.2012.04.025. |
[5] |
H. Brezis and E. Lieb, A relation between pointwise conergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983) 486-490.
doi: 10.1090/S0002-9939-1983-0699419-3. |
[6] |
C. O. Alves, F. J. S. A. Corra and T. F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl., 49 (2005), 85-93.
doi: 10.1016/j.camwa.2005.01.008. |
[7] |
C. T. Cheng and X. Wu, Existence results of positive solutions of Kirchhoff type problems, Nonlinear Anal., 71 (2009), 4883-4892.
doi: 10.1016/j.na.2009.03.065. |
[8] |
C. O. Alves, F. J. S. A. Corra and G. M. Figueiredo, On a class of nonlocal elliptic problems with critical growth, Differential Equation and Applications, 23 (2010), 409-417. |
[9] |
H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983) 437-477.
doi: 10.1002/cpa.3160360405. |
[10] |
J. H. Jin and X. Wu, Infinitely many radial solutions for Kirchhoff-type problems in $R^3$, J. Math. Anal. Appl., 369 (2010). 564-574.
doi: 10.1016/j.jmaa.2010.03.059. |
[11] |
J. J. Nie and X. Wu, Existence and multiplicity of non-trivial solutions for Schrödinger-Kirchhoff-type equations with radial potential, Nonlinear Anal., 75 (2012), 3470-3479.
doi: 10.1016/j.na.2012.01.004. |
[12] |
J. Wang, L. X. Tian, J. X. Xu and F. B. Zhang, Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differential Equations, 253 (2012), 2314-2351.
doi: 10.1016/j.jde.2012.05.023. |
[13] |
J. J. Sun and C. L. Tang, Existence and multipicity of solutions for Kirchhoff type equations, Nonlinear Anal., 74 (2011), 1212-1222.
doi: 10.1016/j.na.2010.09.061. |
[14] |
K. Perera and Z. T. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang-index, J. Differential Equations, 221 (2006), 246-255.
doi: 10.1016/j.jde.2005.03.006,. |
[15] |
L. Wei and X. M. He, Multiplicity of high energy solutions for superlinear Kirchhoff equations, J. Appl. Math. Comput., 39 (2012), 473-487.
doi: 10.1007/s12190-120-0536-1. |
[16] |
P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Application to Differetial Equations," in: CBMS Reg. Conf. Series. Math. 65, Amer. Math. Soc., Providence, RI, 1986. |
[17] |
S. Bernstein, Sur une classe d'équations fonctionnelles aux dérivées, Bull. Acad. Sci. URSS, Sér 17-26 (1940), (Izvestia Akad. Nauk SSSR) 313-345. |
[18] |
S. I. Pohožaev, A certain class of quasilinear hyperbolic equations, Mat. Sb. (NS) 138 (1975), 152-166, 168 (in Russian).
doi: 10.1070/SM1975v025n01ABEH002203. |
[19] |
T. F. Ma and J. E. M. Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem, Appl. Math. Lett., 16 (2003), 243-248.
doi: 10.1016/S0893-9659(03)80038-1,. |
[20] |
X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in $R^3$, Nonlinear Anal. Real World Appl., 12 (2011), 1278-1287.
doi: 10.1016/j.nonrwa.2010.09.023. |
[21] |
X. M. He and W. M. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in $R^3$, J. Differential Equations, 252 (2012), 1813-1834.
doi: 10.1016/j.jde.2011.08.035. |
[22] |
Y. H. Li, F. Y. Li and J. P. Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differential Equations, 253 (2012), 2285-2294.
doi: 10.1016/j.jde.2012.05.017. |
[23] |
Y. Yang and J. H. Zhang, Positive and negative solutions of a class of nonlocal problems, Nonlinear Anal., 73 (2010), 25-30.
doi: 10.1016/j.na.2010.02.008. |
[24] |
Z. T. Zhang and K. Perera, Sign-changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. Math. Anal. Appl., 317 (2006), 456-463.
doi: 10.1016/j.jmaa.2005.06.102. |
show all references
References:
[1] |
A. M. Mao and Z. T. Zhang, Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition, Nonlinear Anal., 70 (2009), 1275-1287.
doi: 10.1016/j.na.2008.02.011. |
[2] |
A. Hamydy, M. Massar and N. Tsouli, Existence of solution for p-Kirchhoff type problems with critical exponents, Electronic J. Differential Equations, 105 (2011), 1-8. |
[3] |
A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 41 (1973), 349-381.
doi: 10.1016/0022-1236(73)90051-7. |
[4] |
B. Cheng, New existence and multiplicity of nontrivial solutions for nonlocal elliptic Kirchhoff type problems, J. Math. Anal. Appl., 394 (2012), 488-495.
doi: 10.1016/j.jmaa.2012.04.025. |
[5] |
H. Brezis and E. Lieb, A relation between pointwise conergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983) 486-490.
doi: 10.1090/S0002-9939-1983-0699419-3. |
[6] |
C. O. Alves, F. J. S. A. Corra and T. F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl., 49 (2005), 85-93.
doi: 10.1016/j.camwa.2005.01.008. |
[7] |
C. T. Cheng and X. Wu, Existence results of positive solutions of Kirchhoff type problems, Nonlinear Anal., 71 (2009), 4883-4892.
doi: 10.1016/j.na.2009.03.065. |
[8] |
C. O. Alves, F. J. S. A. Corra and G. M. Figueiredo, On a class of nonlocal elliptic problems with critical growth, Differential Equation and Applications, 23 (2010), 409-417. |
[9] |
H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983) 437-477.
doi: 10.1002/cpa.3160360405. |
[10] |
J. H. Jin and X. Wu, Infinitely many radial solutions for Kirchhoff-type problems in $R^3$, J. Math. Anal. Appl., 369 (2010). 564-574.
doi: 10.1016/j.jmaa.2010.03.059. |
[11] |
J. J. Nie and X. Wu, Existence and multiplicity of non-trivial solutions for Schrödinger-Kirchhoff-type equations with radial potential, Nonlinear Anal., 75 (2012), 3470-3479.
doi: 10.1016/j.na.2012.01.004. |
[12] |
J. Wang, L. X. Tian, J. X. Xu and F. B. Zhang, Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differential Equations, 253 (2012), 2314-2351.
doi: 10.1016/j.jde.2012.05.023. |
[13] |
J. J. Sun and C. L. Tang, Existence and multipicity of solutions for Kirchhoff type equations, Nonlinear Anal., 74 (2011), 1212-1222.
doi: 10.1016/j.na.2010.09.061. |
[14] |
K. Perera and Z. T. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang-index, J. Differential Equations, 221 (2006), 246-255.
doi: 10.1016/j.jde.2005.03.006,. |
[15] |
L. Wei and X. M. He, Multiplicity of high energy solutions for superlinear Kirchhoff equations, J. Appl. Math. Comput., 39 (2012), 473-487.
doi: 10.1007/s12190-120-0536-1. |
[16] |
P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Application to Differetial Equations," in: CBMS Reg. Conf. Series. Math. 65, Amer. Math. Soc., Providence, RI, 1986. |
[17] |
S. Bernstein, Sur une classe d'équations fonctionnelles aux dérivées, Bull. Acad. Sci. URSS, Sér 17-26 (1940), (Izvestia Akad. Nauk SSSR) 313-345. |
[18] |
S. I. Pohožaev, A certain class of quasilinear hyperbolic equations, Mat. Sb. (NS) 138 (1975), 152-166, 168 (in Russian).
doi: 10.1070/SM1975v025n01ABEH002203. |
[19] |
T. F. Ma and J. E. M. Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem, Appl. Math. Lett., 16 (2003), 243-248.
doi: 10.1016/S0893-9659(03)80038-1,. |
[20] |
X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in $R^3$, Nonlinear Anal. Real World Appl., 12 (2011), 1278-1287.
doi: 10.1016/j.nonrwa.2010.09.023. |
[21] |
X. M. He and W. M. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in $R^3$, J. Differential Equations, 252 (2012), 1813-1834.
doi: 10.1016/j.jde.2011.08.035. |
[22] |
Y. H. Li, F. Y. Li and J. P. Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differential Equations, 253 (2012), 2285-2294.
doi: 10.1016/j.jde.2012.05.017. |
[23] |
Y. Yang and J. H. Zhang, Positive and negative solutions of a class of nonlocal problems, Nonlinear Anal., 73 (2010), 25-30.
doi: 10.1016/j.na.2010.02.008. |
[24] |
Z. T. Zhang and K. Perera, Sign-changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. Math. Anal. Appl., 317 (2006), 456-463.
doi: 10.1016/j.jmaa.2005.06.102. |
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