-
Previous Article
Bifurcation analysis of bursting solutions of two Hindmarsh-Rose neurons with joint electrical and synaptic coupling
- DCDS-B Home
- This Issue
-
Next Article
Delay-induced synchronization transition in small-world Hodgkin-Huxley neuronal networks with channel blocking
Positive solutions of $p$-Laplacian equations with nonlinear boundary condition
1. | Institute of Mathematics, School of Mathematics Science, Nanjing Normal University, Jiangsu Nanjing 210046, China, China |
2. | School of Mathematics and Physics, Xuzhou Institute of Technology, Xuzhou, Jiangsu 221111, China |
References:
[1] |
P. Amster, M. C. Mariani and O. Mendez, Nonlinear boundary conditions for elliptic equations, Electronic Journal of Differential Equations, 144 (2005), 1-8. |
[2] |
J. F. Bonder and J. D. Rossi, Existence Results for the p-Laplacian with nonlinear boundary conditions, J. M. A. A., 263 (2001), 195-223. |
[3] |
J. F. Bonder, J. P. Pinasco and J. D. Rossi, Existence results for a Hamiltonian elliptic systems with nonlinear boundary conditions, Electronic Journal of Differential Equations, 40 (1999), 1-15. |
[4] |
J. F. Bonder and J. D. Rossi, Existence for an elliptic system with nonlinear boundary conditions via fixed point methods, Adv.Differential Equations, 6 (2001), 1-20. |
[5] |
M. Chipot, I. Shafrir and M. Fila, On the solutions to some elliptic equations with nonlinear boundary conditions, Adv. Differential Equations, 1 (1996), 91-110. |
[6] |
M. Chipot, M. Chlebik, M. Fila and I. Shafrir, Existence of positive solutions of a semilinear elliptic equation in $R$N+with a nonlinear boundary condition, J. Math. Anal. Appl., 223 (1998), 429-471.
doi: 10.1006/jmaa.1998.5958. |
[7] |
P. Drabek, Nonlinear eigenvalue problems for the p.Laplacian in $R$N, Math. Nachr., 173 (1995), 131-139.
doi: 10.1002/mana.19951730109. |
[8] |
J. I. Diaz, Nonlinear partial differential equations and free boundaries, Elliptic equations. Pitman Adv. Publ., Boston MA, 323 (1985), 44-95. |
[9] |
P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, Reg. conf. ser. Math, 65 (1986), 1-100. |
[10] |
C. Flores and M. del Pino, Asymptotic behavior of best constants and extremals for trace embeddings in expanding domains, Comm. Partial Differential Equations, 26 (2001), 2189-2210.
doi: 10.1081/PDE-100107818. |
[11] |
Z. M. Guo, Some existence and multiplicity results for a class of quasilinear elliptic eigenvalue problems, Nonlinear Anal., 18 (1992), 957-971.
doi: 10.1016/0362-546X(92)90132-X. |
[12] |
B. Hu, Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary condition, Differential Integral Equations, 7 (1994), 301-313. |
[13] |
Y. ILYasov and T. Runst, Positive solutions for indefinite inhomogeneous Neumann elliptic problems, Electronic Journal of Differential Equations, 57 (2003), 1-21. |
[14] |
D. A. Kandilakis and A. N. Lyberopoulos, Indefinite quasilinear elliptic problems with subcritical and supercritical nonlinearities on unbounded domains, J. Differential Equations, 230 (2006), 337-361.
doi: 10.1016/j.jde.2006.03.008. |
[15] |
E. Montefusco and V. Radulescu, Nonlinear eigenvalue problems for quasilinear operators on unbounded domains, Nonlinear Differ. Equ. Appl., 8 (2001), 481-497.
doi: 10.1007/PL00001460. |
[16] |
K. Pflüger, Existence and multiplicity of solutions to a p-Laplacian equation with nonlinear boundary condition, Electronic Journal of Differential Equations, 10 (1998), 1-13. |
[17] |
R. Filippucci, P. Pucci and v. Rădulescu, Existence and non-existence results for quasilinear elliptic exterior problems with nonlinear boundary conditions, Comm. Partial Differential Equations, 33 (2008), 706-717.
doi: 10.1080/03605300701518208. |
[18] |
X. C. Song, W. H. Wang and P. H. Zhao, Positive solutions of elliptic equations with nonlinear boundary conditions, Nonlinear Analysis, 70 (2009), 328-334.
doi: 10.1016/j.na.2007.12.003. |
[19] |
S. Z. Song and C. L. Tang, Resonance problems for the p-Laplacian with a nonlinear boundary condition, Nonlinear Analysis, 64 (2006), 2007-2021.
doi: 10.1016/j.na.2005.07.035. |
[20] |
S. Terraccini, Symmetry properties of positive solutions to some elliptic equations with nonlinear boundary conditions, Differential Integral Equations, 8 (1995), 1911-1922. |
[21] |
J. H. Zhao and P. H. Zhao, Infinitely many weak solutions for a p-Laplacian equation with nonlinear boundary conditions, EJDE, 90 (2007), 1-14. |
[22] |
J. H. Zhao and P. H. Zhao, Existence of infinitely many weak solutions for the p-Laplacian with nonlinear boundary conditions, Nonlinear Analysis, 69 (2008), 1343-1355.
doi: 10.1016/j.na.2007.06.036. |
[23] |
Z. M. Guo and J. R. L. Webb, Uniqueness of positive solutions for quasilinear elliptic equations when a parameter is large, Proc. Roy. Soc. Edinburgh, 124A (1994), 189-198. |
[24] |
Z. M. Guo, On the number of positive solutions for quasilinear elliptic eigenvalue problems, Nonlinear Analysis, 27 (1996), 229-247.
doi: 10.1016/0362-546X(94)00352-I. |
[25] |
D. D. Hai, Positive solutions of quasilinear boundary value problems, J. Math.Anal. Appl., 217 (1998), 672-686.
doi: 10.1006/jmaa.1997.5762. |
[26] |
B. J. Xuan and Z. C. Chen, Solvability of singular quasilinear elliptic equation, Chinese Ann of Math., 20A (1999), 117-128. |
[27] |
Z. D. Yang and Q. S. Lu, Existence and multipicity of positive entire solutions for a class of quasilinear elliptic equation, Journal Beijing University of Aeronautics and Astronautics, 27 (2001), 217-220. |
[28] |
Z. D. Yang, Existence of positive bounded entire solutions for quasilinear elliptic equations, Appl. Math. Comput., 156 (2004), 743-754.
doi: 10.1016/j.amc.2003.06.024. |
[29] |
M. Guedda and L. Veron, Local and global properties of solutions of quasilinear elliptic equations, J. Diff. Equs., 76 (1988), 159-189.
doi: 10.1016/0022-0396(88)90068-X. |
[30] |
Z. D. Yang, Existence of positive entire solutions for singular and non-singular quasi-linear elliptic equation, J. Comm. Appl. Math., 197 (2006), 355-364.
doi: 10.1016/j.cam.2005.08.027. |
[31] |
B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.
doi: 10.1007/BF01221125. |
[32] |
S. Kichenassamy and J. Smoller, On the existence of radial solutions of quasilinear elliptic equations, Nonlinearity, 3 (1990), 677-694.
doi: 10.1088/0951-7715/3/3/008. |
[33] |
A. Mohammed, Positive solutions of the $p$-Laplace equation with singular nonlinearity, J. Math. Anal. Appl., 352 (2009), 234-245.
doi: 10.1016/j.jmaa.2008.06.018. |
show all references
References:
[1] |
P. Amster, M. C. Mariani and O. Mendez, Nonlinear boundary conditions for elliptic equations, Electronic Journal of Differential Equations, 144 (2005), 1-8. |
[2] |
J. F. Bonder and J. D. Rossi, Existence Results for the p-Laplacian with nonlinear boundary conditions, J. M. A. A., 263 (2001), 195-223. |
[3] |
J. F. Bonder, J. P. Pinasco and J. D. Rossi, Existence results for a Hamiltonian elliptic systems with nonlinear boundary conditions, Electronic Journal of Differential Equations, 40 (1999), 1-15. |
[4] |
J. F. Bonder and J. D. Rossi, Existence for an elliptic system with nonlinear boundary conditions via fixed point methods, Adv.Differential Equations, 6 (2001), 1-20. |
[5] |
M. Chipot, I. Shafrir and M. Fila, On the solutions to some elliptic equations with nonlinear boundary conditions, Adv. Differential Equations, 1 (1996), 91-110. |
[6] |
M. Chipot, M. Chlebik, M. Fila and I. Shafrir, Existence of positive solutions of a semilinear elliptic equation in $R$N+with a nonlinear boundary condition, J. Math. Anal. Appl., 223 (1998), 429-471.
doi: 10.1006/jmaa.1998.5958. |
[7] |
P. Drabek, Nonlinear eigenvalue problems for the p.Laplacian in $R$N, Math. Nachr., 173 (1995), 131-139.
doi: 10.1002/mana.19951730109. |
[8] |
J. I. Diaz, Nonlinear partial differential equations and free boundaries, Elliptic equations. Pitman Adv. Publ., Boston MA, 323 (1985), 44-95. |
[9] |
P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, Reg. conf. ser. Math, 65 (1986), 1-100. |
[10] |
C. Flores and M. del Pino, Asymptotic behavior of best constants and extremals for trace embeddings in expanding domains, Comm. Partial Differential Equations, 26 (2001), 2189-2210.
doi: 10.1081/PDE-100107818. |
[11] |
Z. M. Guo, Some existence and multiplicity results for a class of quasilinear elliptic eigenvalue problems, Nonlinear Anal., 18 (1992), 957-971.
doi: 10.1016/0362-546X(92)90132-X. |
[12] |
B. Hu, Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary condition, Differential Integral Equations, 7 (1994), 301-313. |
[13] |
Y. ILYasov and T. Runst, Positive solutions for indefinite inhomogeneous Neumann elliptic problems, Electronic Journal of Differential Equations, 57 (2003), 1-21. |
[14] |
D. A. Kandilakis and A. N. Lyberopoulos, Indefinite quasilinear elliptic problems with subcritical and supercritical nonlinearities on unbounded domains, J. Differential Equations, 230 (2006), 337-361.
doi: 10.1016/j.jde.2006.03.008. |
[15] |
E. Montefusco and V. Radulescu, Nonlinear eigenvalue problems for quasilinear operators on unbounded domains, Nonlinear Differ. Equ. Appl., 8 (2001), 481-497.
doi: 10.1007/PL00001460. |
[16] |
K. Pflüger, Existence and multiplicity of solutions to a p-Laplacian equation with nonlinear boundary condition, Electronic Journal of Differential Equations, 10 (1998), 1-13. |
[17] |
R. Filippucci, P. Pucci and v. Rădulescu, Existence and non-existence results for quasilinear elliptic exterior problems with nonlinear boundary conditions, Comm. Partial Differential Equations, 33 (2008), 706-717.
doi: 10.1080/03605300701518208. |
[18] |
X. C. Song, W. H. Wang and P. H. Zhao, Positive solutions of elliptic equations with nonlinear boundary conditions, Nonlinear Analysis, 70 (2009), 328-334.
doi: 10.1016/j.na.2007.12.003. |
[19] |
S. Z. Song and C. L. Tang, Resonance problems for the p-Laplacian with a nonlinear boundary condition, Nonlinear Analysis, 64 (2006), 2007-2021.
doi: 10.1016/j.na.2005.07.035. |
[20] |
S. Terraccini, Symmetry properties of positive solutions to some elliptic equations with nonlinear boundary conditions, Differential Integral Equations, 8 (1995), 1911-1922. |
[21] |
J. H. Zhao and P. H. Zhao, Infinitely many weak solutions for a p-Laplacian equation with nonlinear boundary conditions, EJDE, 90 (2007), 1-14. |
[22] |
J. H. Zhao and P. H. Zhao, Existence of infinitely many weak solutions for the p-Laplacian with nonlinear boundary conditions, Nonlinear Analysis, 69 (2008), 1343-1355.
doi: 10.1016/j.na.2007.06.036. |
[23] |
Z. M. Guo and J. R. L. Webb, Uniqueness of positive solutions for quasilinear elliptic equations when a parameter is large, Proc. Roy. Soc. Edinburgh, 124A (1994), 189-198. |
[24] |
Z. M. Guo, On the number of positive solutions for quasilinear elliptic eigenvalue problems, Nonlinear Analysis, 27 (1996), 229-247.
doi: 10.1016/0362-546X(94)00352-I. |
[25] |
D. D. Hai, Positive solutions of quasilinear boundary value problems, J. Math.Anal. Appl., 217 (1998), 672-686.
doi: 10.1006/jmaa.1997.5762. |
[26] |
B. J. Xuan and Z. C. Chen, Solvability of singular quasilinear elliptic equation, Chinese Ann of Math., 20A (1999), 117-128. |
[27] |
Z. D. Yang and Q. S. Lu, Existence and multipicity of positive entire solutions for a class of quasilinear elliptic equation, Journal Beijing University of Aeronautics and Astronautics, 27 (2001), 217-220. |
[28] |
Z. D. Yang, Existence of positive bounded entire solutions for quasilinear elliptic equations, Appl. Math. Comput., 156 (2004), 743-754.
doi: 10.1016/j.amc.2003.06.024. |
[29] |
M. Guedda and L. Veron, Local and global properties of solutions of quasilinear elliptic equations, J. Diff. Equs., 76 (1988), 159-189.
doi: 10.1016/0022-0396(88)90068-X. |
[30] |
Z. D. Yang, Existence of positive entire solutions for singular and non-singular quasi-linear elliptic equation, J. Comm. Appl. Math., 197 (2006), 355-364.
doi: 10.1016/j.cam.2005.08.027. |
[31] |
B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.
doi: 10.1007/BF01221125. |
[32] |
S. Kichenassamy and J. Smoller, On the existence of radial solutions of quasilinear elliptic equations, Nonlinearity, 3 (1990), 677-694.
doi: 10.1088/0951-7715/3/3/008. |
[33] |
A. Mohammed, Positive solutions of the $p$-Laplace equation with singular nonlinearity, J. Math. Anal. Appl., 352 (2009), 234-245.
doi: 10.1016/j.jmaa.2008.06.018. |
[1] |
Dorota Bors. Application of Mountain Pass Theorem to superlinear equations with fractional Laplacian controlled by distributed parameters and boundary data. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 29-43. doi: 10.3934/dcdsb.2018003 |
[2] |
Takashi Kajiwara. The sub-supersolution method for the FitzHugh-Nagumo type reaction-diffusion system with heterogeneity. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2441-2465. doi: 10.3934/dcds.2018101 |
[3] |
Christina A. Hollon, Jeffrey T. Neugebauer. Positive solutions of a fractional boundary value problem with a fractional derivative boundary condition. Conference Publications, 2015, 2015 (special) : 615-620. doi: 10.3934/proc.2015.0615 |
[4] |
John V. Baxley, Philip T. Carroll. Nonlinear boundary value problems with multiple positive solutions. Conference Publications, 2003, 2003 (Special) : 83-90. doi: 10.3934/proc.2003.2003.83 |
[5] |
Patricio Cerda, Leonelo Iturriaga, Sebastián Lorca, Pedro Ubilla. Positive radial solutions of a nonlinear boundary value problem. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1765-1783. doi: 10.3934/cpaa.2018084 |
[6] |
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 |
[7] |
John R. Graef, Lingju Kong, Bo Yang. Positive solutions of a nonlinear higher order boundary-value problem. Conference Publications, 2009, 2009 (Special) : 276-285. doi: 10.3934/proc.2009.2009.276 |
[8] |
Eric R. Kaufmann. Existence and nonexistence of positive solutions for a nonlinear fractional boundary value problem. Conference Publications, 2009, 2009 (Special) : 416-423. doi: 10.3934/proc.2009.2009.416 |
[9] |
Dmitry Glotov, P. J. McKenna. Numerical mountain pass solutions of Ginzburg-Landau type equations. Communications on Pure and Applied Analysis, 2008, 7 (6) : 1345-1359. doi: 10.3934/cpaa.2008.7.1345 |
[10] |
Claudianor O. Alves, Giovany M. Figueiredo, Marcelo F. Furtado. Multiplicity of solutions for elliptic systems via local Mountain Pass method. Communications on Pure and Applied Analysis, 2009, 8 (6) : 1745-1758. doi: 10.3934/cpaa.2009.8.1745 |
[11] |
Christopher Grumiau, Marco Squassina, Christophe Troestler. On the Mountain-Pass algorithm for the quasi-linear Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1345-1360. doi: 10.3934/dcdsb.2013.18.1345 |
[12] |
VicenŢiu D. RǍdulescu, Somayeh Saiedinezhad. A nonlinear eigenvalue problem with $ p(x) $-growth and generalized Robin boundary value condition. Communications on Pure and Applied Analysis, 2018, 17 (1) : 39-52. doi: 10.3934/cpaa.2018003 |
[13] |
Tsung-Fang Wu. Multiplicity of positive solutions for a semilinear elliptic equation in $R_+^N$ with nonlinear boundary condition. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1675-1696. doi: 10.3934/cpaa.2010.9.1675 |
[14] |
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 and Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825 |
[15] |
Michael E. Filippakis, Nikolaos S. Papageorgiou. Existence and multiplicity of positive solutions for nonlinear boundary value problems driven by the scalar $p$-Laplacian. Communications on Pure and Applied Analysis, 2004, 3 (4) : 729-756. doi: 10.3934/cpaa.2004.3.729 |
[16] |
Johnny Henderson, Rodica Luca. Existence of positive solutions for a system of nonlinear second-order integral boundary value problems. Conference Publications, 2015, 2015 (special) : 596-604. doi: 10.3934/proc.2015.0596 |
[17] |
Hongjing Pan, Ruixiang Xing. On the existence of positive solutions for some nonlinear boundary value problems and applications to MEMS models. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3627-3682. doi: 10.3934/dcds.2015.35.3627 |
[18] |
Xuemei Zhang, Meiqiang Feng. Double bifurcation diagrams and four positive solutions of nonlinear boundary value problems via time maps. Communications on Pure and Applied Analysis, 2018, 17 (5) : 2149-2171. doi: 10.3934/cpaa.2018103 |
[19] |
G. Infante. Positive solutions of nonlocal boundary value problems with singularities. Conference Publications, 2009, 2009 (Special) : 377-384. doi: 10.3934/proc.2009.2009.377 |
[20] |
John R. Graef, Lingju Kong, Qingkai Kong, Min Wang. Positive solutions of nonlocal fractional boundary value problems. Conference Publications, 2013, 2013 (special) : 283-290. doi: 10.3934/proc.2013.2013.283 |
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]