-
Previous Article
Convergence rates for elliptic reiterated homogenization problems
- CPAA Home
- This Issue
-
Next Article
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.
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.
|
[3] |
A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications,, J. Funct. Anal., 41 (1973), 349.
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.
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.
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.
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.
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.
|
[9] |
H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,, Comm. Pure Appl. Math., 36 (1983), 437.
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.
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.
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.
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.
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.
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.
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, (1986).
|
[17] |
S. Bernstein, Sur une classe d'équations fonctionnelles aux dérivées,, Bull. Acad. Sci. URSS, 17-26 (1940), 17.
|
[18] |
S. I. Pohožaev, A certain class of quasilinear hyperbolic equations,, Mat. Sb. (NS) \textbf{138} (1975), 138 (1975), 152.
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.
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.
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.
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.
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.
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.
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.
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.
|
[3] |
A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications,, J. Funct. Anal., 41 (1973), 349.
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.
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.
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.
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.
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.
|
[9] |
H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,, Comm. Pure Appl. Math., 36 (1983), 437.
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.
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.
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.
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.
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.
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.
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, (1986).
|
[17] |
S. Bernstein, Sur une classe d'équations fonctionnelles aux dérivées,, Bull. Acad. Sci. URSS, 17-26 (1940), 17.
|
[18] |
S. I. Pohožaev, A certain class of quasilinear hyperbolic equations,, Mat. Sb. (NS) \textbf{138} (1975), 138 (1975), 152.
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.
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.
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.
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.
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.
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.
doi: 10.1016/j.jmaa.2005.06.102. |
[1] |
Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309 |
[2] |
Alexandr Mikhaylov, Victor Mikhaylov. Dynamic inverse problem for Jacobi matrices. Inverse Problems & Imaging, 2019, 13 (3) : 431-447. doi: 10.3934/ipi.2019021 |
[3] |
Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271 |
[4] |
Hildeberto E. Cabral, Zhihong Xia. Subharmonic solutions in the restricted three-body problem. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 463-474. doi: 10.3934/dcds.1995.1.463 |
[5] |
Michel Chipot, Mingmin Zhang. On some model problem for the propagation of interacting species in a special environment. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020401 |
[6] |
Fritz Gesztesy, Helge Holden, Johanna Michor, Gerald Teschl. The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 151-196. doi: 10.3934/dcds.2010.26.151 |
[7] |
Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261 |
[8] |
Rongchang Liu, Jiangyuan Li, Duokui Yan. New periodic orbits in the planar equal-mass three-body problem. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2187-2206. doi: 10.3934/dcds.2018090 |
[9] |
Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056 |
[10] |
Z. Reichstein and B. Youssin. Parusinski's "Key Lemma" via algebraic geometry. Electronic Research Announcements, 1999, 5: 136-145. |
[11] |
Madalina Petcu, Roger Temam. The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 209-222. doi: 10.3934/dcdss.2011.4.209 |
[12] |
Martial Agueh, Reinhard Illner, Ashlin Richardson. Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type. Kinetic & Related Models, 2011, 4 (1) : 1-16. doi: 10.3934/krm.2011.4.1 |
[13] |
Andrea Cianchi, Adele Ferone. Improving sharp Sobolev type inequalities by optimal remainder gradient norms. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1363-1386. doi: 10.3934/cpaa.2012.11.1363 |
[14] |
Fernando P. da Costa, João T. Pinto, Rafael Sasportes. On the convergence to critical scaling profiles in submonolayer deposition models. Kinetic & Related Models, 2018, 11 (6) : 1359-1376. doi: 10.3934/krm.2018053 |
[15] |
Gioconda Moscariello, Antonia Passarelli di Napoli, Carlo Sbordone. Planar ACL-homeomorphisms : Critical points of their components. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1391-1397. doi: 10.3934/cpaa.2010.9.1391 |
[16] |
Ian Schindler, Kyril Tintarev. Mountain pass solutions to semilinear problems with critical nonlinearity. Conference Publications, 2007, 2007 (Special) : 912-919. doi: 10.3934/proc.2007.2007.912 |
[17] |
Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825 |
[18] |
Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037 |
[19] |
Yuncherl Choi, Taeyoung Ha, Jongmin Han, Sewoong Kim, Doo Seok Lee. Turing instability and dynamic phase transition for the Brusselator model with multiple critical eigenvalues. Discrete & Continuous Dynamical Systems - A, 2021 doi: 10.3934/dcds.2021035 |
[20] |
Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533 |
2019 Impact Factor: 1.105
Tools
Metrics
Other articles
by authors
[Back to Top]