Kirchhoff type equations with strong singularities

Yijing Sun; Yuxin Tan

doi:10.3934/cpaa.2019010

Previous Article
Existence of weak solutions for a diffuse interface model for two-phase flow with surfactants
CPAA Home
This Issue
Next Article
Existence and a general decay results for a viscoelastic plate equation with a logarithmic nonlinearity

January 2019, 18(1): 181-193. doi: 10.3934/cpaa.2019010

Kirchhoff type equations with strong singularities

Yijing Sun ^, and Yuxin Tan

Department of Mathematics, University of Chinese Academy of Sciences, Beijing 100049, China

* Corresponding author

Received October 2017 Revised January 2018 Published August 2018

Fund Project: The authors are supported by NSFC grants 11571339 and 11771468.

An optimal condition is given for the existence of positive solutions of nonlinear Kirchhoff PDE with strong singularities. A byproduct is that $-2$ is no longer the critical position for the existence of positive solutions of PDE's with singular potentials and negative powers of the form: $ - |x{|^\alpha }\Delta u = {u^{{\rm{ - }}\gamma }}$ in $Ω$, $u = 0$ on $\partial \Omega $, where $\Omega$ is a bounded domain of ${\mathbb{R}}^{N}$ containing 0, with $N \ge 3$, $\alpha \in \left( {0, N} \right)$ and $ - \gamma \in \left( { - 3, - 1} \right)$.

Keywords: Kirchhoff type equations, strong singularity, singular potential.

Mathematics Subject Classification: Primary: 35J70, 35J20; Secondary: 35J60.

Citation: Yijing Sun, Yuxin Tan. Kirchhoff type equations with strong singularities. Communications on Pure & Applied Analysis, 2019, 18 (1) : 181-193. doi: 10.3934/cpaa.2019010

References:

[1]	R. P. Agarwal and D. O'Regan, Singular Differential and Integral Equations with Applications, Springer, New York, 2003. doi: 10.1007/978-94-017-3004-4. Google Scholar
[2]	C. Alves, F. Correa and J. Goncalves, Existence of solutions for some classes of singular Hamiltonian systems, Advanced Nonlinear Studies, 5 (2005), 265-278. doi: 10.1515/ans-2005-0206. Google Scholar
[3]	C. Alves and M. Montenegro, Positive solutions to a singular Neumann problem, J. Math. Anal. Appl., 352 (2009), 112-119. doi: 10.1016/j.jmaa.2008.02.026. Google Scholar
[4]	L. Bai and G. Zhang, Nontrivial solutions for a nonlinear discrete elliptic equation with periodic boundary conditions, Appl. Math. Comput., 210 (2009), 321-333. doi: 10.1016/j.amc.2008.12.024. Google Scholar
[5]	L. Boccardo and L. Orsina, Semilinear elliptic equations with singular nonlinearities, Calc. Var., 37 (2010), 636-380. doi: 10.1007/s00526-009-0266-x. Google Scholar
[6]	P. Caldiroli and R. Musina, On a class of two-dimensional singular elliptic problems, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 479-497. doi: 10.1017/S0308210500000974. Google Scholar
[7]	J. Chabrowski, On the Neumann problem with singular and superlinear nonlinearities, Comm. in Applied Analysis, 13 (2009), 327-340. Google Scholar
[8]	M. Chhetri, S. Raynor and S. Rabinson, On the existence of multiple positive solutions to some superlinear systems, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 39-59. doi: 10.1017/S0308210510000582. Google Scholar
[9]	M. Coclite and G. Palmieri, On a singular nonlinear Dirichlet problem, Comm. Partial Differential Equations, 14 (1989), 1315-1327. doi: 10.1080/03605308908820656. Google Scholar
[10]	M. Crandall, P. Rabinowitz and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations, 2 (1977), 2615-2622. doi: 10.1080/03605307708820029. Google Scholar
[11]	J. I. Diaz, J. Hernández and J. M. Rakotoson, On very weak positive solutions to some semilinear elliptic problems with simultaneous singular nonlinear and spatial dependence terms, Milan J. Math., 79 (2011), 233-245. doi: 10.1007/s00032-011-0151-x. Google Scholar
[12]	J. I. Diaz, J. Morel and L. Oswald, An elliptic equation with singular nonlinearity, Comm. in Partial Differential Equations, 12 (1987), 1333-1344. doi: 10.1080/03605308708820531. Google Scholar
[13]	I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353. doi: 10.1016/0022-247X(74)90025-0. Google Scholar
[14]	L. Gasinski and N. Papageorgiou, Nonlinear elliptic equations with singular terms and combined nonlinearities, Annales Henri Poincare, 13 (2012), 481-512. doi: 10.1007/s00023-011-0129-9. Google Scholar
[15]	M. Ghergu and V. Radulescu, Ground state solutions for the singular Lane-Emden-Fowler equation with sublinear convection term, J. Math. Anal. Appl., 333 (2007), 265-273. doi: 10.1016/j.jmaa.2006.09.074. Google Scholar
[16]	J. Giacomoni, S. Prashanth and K. Sreenadh, Uniqueness and multiplicity results for N-Laplace equation with critical and singular nonlinearity in a ball, Asymptotic Analysis, 61 (2009), 195-227. Google Scholar
[17]	J. Giacomoni and K. Saoudi, Multiplicity of positive solutions for a singular and critical problem, Nonlinear Anal., 71 (2009), 4060-4077. doi: 10.1016/j.na.2009.02.087. Google Scholar
[18]	J. Giacomoni, I. Schindler and P. Takac, Sobolev versus Holder local minimizers and existence of multiple solutions for a singular quasilinear equation, Annali Della Scuola Norm. Sup. Pisa, 6 (2007), 117-158. Google Scholar
[19]	J. Giacomoni and K. Sreenadh, Multiplicity results for a singular and quasilinear equation, Discrete Continuous Dynamical Systems, (2007), 429-435. Google Scholar
[20]	J. Goncalves, A. Melo and C. Santos, On existence of L-infinity-gound states for singular elliptic equations in the presence of a strongly nonlinear term, Advanced Nonlinear Studies, 7 (2007), 475-490. doi: 10.1515/ans-2007-0308. Google Scholar
[21]	J. Goncalves and C. Santos, Singular ellitptic problems: Existence, non-existence and boundary behavior, Nonlinear Anal., 66 (2007), 2078-2090. doi: 10.1016/j.na.2006.03.003. Google Scholar
[22]	C. F. Gui and F. H. Lin, Regularity of an elliptic problem with a singular nonlinearity, Proc. Roy. Soc. Edinburgh, 123A (1993), 1021-1029. doi: 10.1017/S030821050002970X. Google Scholar
[23]	D. Hai, On an asymptotically linear singular boundary value problems, Topological Methods in Nonlinear Analysis, 39 (2012), 83-92. Google Scholar
[24]	J. Hernández, F. J. Mancebo and J. M. Vega, Positive solutions for singular nonlinear elliptic equations, Proc. Roy. Soc. Edinburgh, 137A (2007), 41-62. doi: 10.1017/S030821050500065X. Google Scholar
[25]	J. Hernández and F. J. Mancebo, Singular elliptic and parabolic equations, in Handbook of Differential Equations: Stationary Partial Differential Equations, Vol. 3, Elsevier, 317-400, (2006) Google Scholar
[26]	N. Hirano, C. Saccon and N. Shioji, Brezis-Nirenberg type theorems and multiplicity of positive solutions for a singular elliptic problem, J. Differential Equations, 245 (2008), 1997-2037. doi: 10.1016/j.jde.2008.06.020. Google Scholar
[27]	G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. Google Scholar
[28]	S. Kyritsi and N. Papageorgiou, Pairs of positive solutions for singular p-Laplacian equations with a p-superlinear potential, Nonlinear Anal., 73 (2010), 1136-1142. doi: 10.1016/j.na.2010.04.019. Google Scholar
[29]	A. V. Lair and A. W. Shaker, Classical and weak solutions of a singular semilinear elliptic problem, J. Math. Anal. Appl., 211 (1997), 371-385. doi: 10.1006/jmaa.1997.5470. Google Scholar
[30]	A. C. Lazer and P. J. McKenna, On a singular nonlinear elliptic boundary value problem, Proc. Amer. Math. Soc., 111 (1991), 721-730. doi: 10.2307/2048410. Google Scholar
[31]	J. F. Liao, X. F. Ke, C. Y. Lei and C. L. Tang, A uniqueness result for Kirchhoff type problems with singularity, Appl.Math.Lett., 59 (2016), 24-30. doi: 10.1016/j.aml.2016.03.001. Google Scholar
[32]	J. L. Lions, On some questions in boundary value problems of mathematical physics, North-Holland Math. Stud, Vol. 30, North-Holland, Amsterdam, 1978, pp. 284-346. Google Scholar
[33]	N. Loc and K. Schmitt, Boundary value problems for singular elliptic equations, Rocky Mountain Journal of Mathematics, 41 (2011), 555-572. doi: 10.1216/RMJ-2011-41-2-555. Google Scholar
[34]	M. Montenegro and E. Silva, Two solutions for s singular elliptic equation by variational methods, Annali Della Scuola Normale Superiore Di Pisa, 11 (2012), 143-165. Google Scholar
[35]	K. Perera and Z. 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. Google Scholar
[36]	J. P. Shi and M. X. Yao, On a singular semilinear elliptic problem, Proc. Roy. Soc. Edinburgh, 128A (1998), 1389-1401. doi: 10.1017/S0308210500027384. Google Scholar
[37]	L. Xing and S. Yijing, Multiple positive solutions for Kirchhoff type problems with singularity, Comm. Pure Appl. Anal., 12 (2013), 721-733. Google Scholar
[38]	S. Yijing, Compatibility phenomena in singular problems, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 1321-1330. doi: 10.1017/S030821051100117X. Google Scholar
[39]	S. Yijing and Z. Duanzhi, The role of the power 3 for elliptic equations with negative exponents, Calc.Var. Partial Differential Equations, 49 (2014), 909-922. doi: 10.1007/s00526-013-0604-x. Google Scholar
[40]	S. Yijing and W. Shaoping, An exact estimate result for a class of singular equations with critical exponents, J. Funct. Anal., 260 (2011), 1257-1284. doi: 10.1016/j.jfa.2010.11.018. Google Scholar
[41]	S. Yijing and L. Yiming, The planar Orlicz Minkowski problem in the L¹-sense, Adv. Math., 281 (2015), 1364-1383. doi: 10.1016/j.aim.2015.03.032. Google Scholar
[42]	Z. Zhang and J. Cheng, Existence and optimal estimates of solutions for singular nonlinear Dirichlet problems, Nonlinear Anal., 57 (2004), 473-484. doi: 10.1016/j.na.2004.02.025. Google Scholar
[43]	Z. 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. Google Scholar

show all references

References:

[1]	R. P. Agarwal and D. O'Regan, Singular Differential and Integral Equations with Applications, Springer, New York, 2003. doi: 10.1007/978-94-017-3004-4. Google Scholar
[2]	C. Alves, F. Correa and J. Goncalves, Existence of solutions for some classes of singular Hamiltonian systems, Advanced Nonlinear Studies, 5 (2005), 265-278. doi: 10.1515/ans-2005-0206. Google Scholar
[3]	C. Alves and M. Montenegro, Positive solutions to a singular Neumann problem, J. Math. Anal. Appl., 352 (2009), 112-119. doi: 10.1016/j.jmaa.2008.02.026. Google Scholar
[4]	L. Bai and G. Zhang, Nontrivial solutions for a nonlinear discrete elliptic equation with periodic boundary conditions, Appl. Math. Comput., 210 (2009), 321-333. doi: 10.1016/j.amc.2008.12.024. Google Scholar
[5]	L. Boccardo and L. Orsina, Semilinear elliptic equations with singular nonlinearities, Calc. Var., 37 (2010), 636-380. doi: 10.1007/s00526-009-0266-x. Google Scholar
[6]	P. Caldiroli and R. Musina, On a class of two-dimensional singular elliptic problems, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 479-497. doi: 10.1017/S0308210500000974. Google Scholar
[7]	J. Chabrowski, On the Neumann problem with singular and superlinear nonlinearities, Comm. in Applied Analysis, 13 (2009), 327-340. Google Scholar
[8]	M. Chhetri, S. Raynor and S. Rabinson, On the existence of multiple positive solutions to some superlinear systems, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 39-59. doi: 10.1017/S0308210510000582. Google Scholar
[9]	M. Coclite and G. Palmieri, On a singular nonlinear Dirichlet problem, Comm. Partial Differential Equations, 14 (1989), 1315-1327. doi: 10.1080/03605308908820656. Google Scholar
[10]	M. Crandall, P. Rabinowitz and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations, 2 (1977), 2615-2622. doi: 10.1080/03605307708820029. Google Scholar
[11]	J. I. Diaz, J. Hernández and J. M. Rakotoson, On very weak positive solutions to some semilinear elliptic problems with simultaneous singular nonlinear and spatial dependence terms, Milan J. Math., 79 (2011), 233-245. doi: 10.1007/s00032-011-0151-x. Google Scholar
[12]	J. I. Diaz, J. Morel and L. Oswald, An elliptic equation with singular nonlinearity, Comm. in Partial Differential Equations, 12 (1987), 1333-1344. doi: 10.1080/03605308708820531. Google Scholar
[13]	I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353. doi: 10.1016/0022-247X(74)90025-0. Google Scholar
[14]	L. Gasinski and N. Papageorgiou, Nonlinear elliptic equations with singular terms and combined nonlinearities, Annales Henri Poincare, 13 (2012), 481-512. doi: 10.1007/s00023-011-0129-9. Google Scholar
[15]	M. Ghergu and V. Radulescu, Ground state solutions for the singular Lane-Emden-Fowler equation with sublinear convection term, J. Math. Anal. Appl., 333 (2007), 265-273. doi: 10.1016/j.jmaa.2006.09.074. Google Scholar
[16]	J. Giacomoni, S. Prashanth and K. Sreenadh, Uniqueness and multiplicity results for N-Laplace equation with critical and singular nonlinearity in a ball, Asymptotic Analysis, 61 (2009), 195-227. Google Scholar
[17]	J. Giacomoni and K. Saoudi, Multiplicity of positive solutions for a singular and critical problem, Nonlinear Anal., 71 (2009), 4060-4077. doi: 10.1016/j.na.2009.02.087. Google Scholar
[18]	J. Giacomoni, I. Schindler and P. Takac, Sobolev versus Holder local minimizers and existence of multiple solutions for a singular quasilinear equation, Annali Della Scuola Norm. Sup. Pisa, 6 (2007), 117-158. Google Scholar
[19]	J. Giacomoni and K. Sreenadh, Multiplicity results for a singular and quasilinear equation, Discrete Continuous Dynamical Systems, (2007), 429-435. Google Scholar
[20]	J. Goncalves, A. Melo and C. Santos, On existence of L-infinity-gound states for singular elliptic equations in the presence of a strongly nonlinear term, Advanced Nonlinear Studies, 7 (2007), 475-490. doi: 10.1515/ans-2007-0308. Google Scholar
[21]	J. Goncalves and C. Santos, Singular ellitptic problems: Existence, non-existence and boundary behavior, Nonlinear Anal., 66 (2007), 2078-2090. doi: 10.1016/j.na.2006.03.003. Google Scholar
[22]	C. F. Gui and F. H. Lin, Regularity of an elliptic problem with a singular nonlinearity, Proc. Roy. Soc. Edinburgh, 123A (1993), 1021-1029. doi: 10.1017/S030821050002970X. Google Scholar
[23]	D. Hai, On an asymptotically linear singular boundary value problems, Topological Methods in Nonlinear Analysis, 39 (2012), 83-92. Google Scholar
[24]	J. Hernández, F. J. Mancebo and J. M. Vega, Positive solutions for singular nonlinear elliptic equations, Proc. Roy. Soc. Edinburgh, 137A (2007), 41-62. doi: 10.1017/S030821050500065X. Google Scholar
[25]	J. Hernández and F. J. Mancebo, Singular elliptic and parabolic equations, in Handbook of Differential Equations: Stationary Partial Differential Equations, Vol. 3, Elsevier, 317-400, (2006) Google Scholar
[26]	N. Hirano, C. Saccon and N. Shioji, Brezis-Nirenberg type theorems and multiplicity of positive solutions for a singular elliptic problem, J. Differential Equations, 245 (2008), 1997-2037. doi: 10.1016/j.jde.2008.06.020. Google Scholar
[27]	G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. Google Scholar
[28]	S. Kyritsi and N. Papageorgiou, Pairs of positive solutions for singular p-Laplacian equations with a p-superlinear potential, Nonlinear Anal., 73 (2010), 1136-1142. doi: 10.1016/j.na.2010.04.019. Google Scholar
[29]	A. V. Lair and A. W. Shaker, Classical and weak solutions of a singular semilinear elliptic problem, J. Math. Anal. Appl., 211 (1997), 371-385. doi: 10.1006/jmaa.1997.5470. Google Scholar
[30]	A. C. Lazer and P. J. McKenna, On a singular nonlinear elliptic boundary value problem, Proc. Amer. Math. Soc., 111 (1991), 721-730. doi: 10.2307/2048410. Google Scholar
[31]	J. F. Liao, X. F. Ke, C. Y. Lei and C. L. Tang, A uniqueness result for Kirchhoff type problems with singularity, Appl.Math.Lett., 59 (2016), 24-30. doi: 10.1016/j.aml.2016.03.001. Google Scholar
[32]	J. L. Lions, On some questions in boundary value problems of mathematical physics, North-Holland Math. Stud, Vol. 30, North-Holland, Amsterdam, 1978, pp. 284-346. Google Scholar
[33]	N. Loc and K. Schmitt, Boundary value problems for singular elliptic equations, Rocky Mountain Journal of Mathematics, 41 (2011), 555-572. doi: 10.1216/RMJ-2011-41-2-555. Google Scholar
[34]	M. Montenegro and E. Silva, Two solutions for s singular elliptic equation by variational methods, Annali Della Scuola Normale Superiore Di Pisa, 11 (2012), 143-165. Google Scholar
[35]	K. Perera and Z. 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. Google Scholar
[36]	J. P. Shi and M. X. Yao, On a singular semilinear elliptic problem, Proc. Roy. Soc. Edinburgh, 128A (1998), 1389-1401. doi: 10.1017/S0308210500027384. Google Scholar
[37]	L. Xing and S. Yijing, Multiple positive solutions for Kirchhoff type problems with singularity, Comm. Pure Appl. Anal., 12 (2013), 721-733. Google Scholar
[38]	S. Yijing, Compatibility phenomena in singular problems, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 1321-1330. doi: 10.1017/S030821051100117X. Google Scholar
[39]	S. Yijing and Z. Duanzhi, The role of the power 3 for elliptic equations with negative exponents, Calc.Var. Partial Differential Equations, 49 (2014), 909-922. doi: 10.1007/s00526-013-0604-x. Google Scholar
[40]	S. Yijing and W. Shaoping, An exact estimate result for a class of singular equations with critical exponents, J. Funct. Anal., 260 (2011), 1257-1284. doi: 10.1016/j.jfa.2010.11.018. Google Scholar
[41]	S. Yijing and L. Yiming, The planar Orlicz Minkowski problem in the L¹-sense, Adv. Math., 281 (2015), 1364-1383. doi: 10.1016/j.aim.2015.03.032. Google Scholar
[42]	Z. Zhang and J. Cheng, Existence and optimal estimates of solutions for singular nonlinear Dirichlet problems, Nonlinear Anal., 57 (2004), 473-484. doi: 10.1016/j.na.2004.02.025. Google Scholar
[43]	Z. 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. Google Scholar

[1]	Pawan Kumar Mishra, Sarika Goyal, K. Sreenadh. Polyharmonic Kirchhoff type equations with singular exponential nonlinearities. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1689-1717. doi: 10.3934/cpaa.2016009
[2]	Yu Chen, Yanheng Ding, Suhong Li. Existence and concentration for Kirchhoff type equations around topologically critical points of the potential. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1641-1671. doi: 10.3934/cpaa.2017079
[3]	Xing Liu, Yijing Sun. Multiple positive solutions for Kirchhoff type problems with singularity. Communications on Pure & Applied Analysis, 2013, 12 (2) : 721-733. doi: 10.3934/cpaa.2013.12.721
[4]	Jijiang Sun, Chun-Lei Tang. Resonance problems for Kirchhoff type equations. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 2139-2154. doi: 10.3934/dcds.2013.33.2139
[5]	Helin Guo, Yimin Zhang, Huansong Zhou. Blow-up solutions for a Kirchhoff type elliptic equation with trapping potential. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1875-1897. doi: 10.3934/cpaa.2018089
[6]	Rui-Qi Liu, Chun-Lei Tang, Jia-Feng Liao, Xing-Ping Wu. Positive solutions of Kirchhoff type problem with singular and critical nonlinearities in dimension four. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1841-1856. doi: 10.3934/cpaa.2016006
[7]	Veronica Felli, Elsa M. Marchini, Susanna Terracini. On the behavior of solutions to Schrödinger equations with dipole type potentials near the singularity. Discrete & Continuous Dynamical Systems, 2008, 21 (1) : 91-119. doi: 10.3934/dcds.2008.21.91
[8]	Mingqi Xiang, Binlin Zhang, Die Hu. Kirchhoff-type differential inclusion problems involving the fractional Laplacian and strong damping. Electronic Research Archive, 2020, 28 (2) : 651-669. doi: 10.3934/era.2020034
[9]	Yahui Niu. A Hopf type lemma and the symmetry of solutions for a class of Kirchhoff equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021027
[10]	Jiu Liu, Jia-Feng Liao, Chun-Lei Tang. Positive solution for the Kirchhoff-type equations involving general subcritical growth. Communications on Pure & Applied Analysis, 2016, 15 (2) : 445-455. doi: 10.3934/cpaa.2016.15.445
[11]	Norihisa Ikoma. Existence of ground state solutions to the nonlinear Kirchhoff type equations with potentials. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 943-966. doi: 10.3934/dcds.2015.35.943
[12]	Quanqing Li, Kaimin Teng, Xian Wu. Ground states for Kirchhoff-type equations with critical growth. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2623-2638. doi: 10.3934/cpaa.2018124
[13]	Ling Ding, Shu-Ming Sun. Existence of positive solutions for a class of Kirchhoff type equations in $\mathbb{R}^3$. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1663-1685. doi: 10.3934/dcdss.2016069
[14]	Peng Chen, Xiaochun Liu. Multiplicity of solutions to Kirchhoff type equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2018, 17 (1) : 113-125. doi: 10.3934/cpaa.2018007
[15]	Jijiang Sun, Shiwang Ma. Nontrivial solutions for Kirchhoff type equations via Morse theory. Communications on Pure & Applied Analysis, 2014, 13 (2) : 483-494. doi: 10.3934/cpaa.2014.13.483
[16]	Yinbin Deng, Wentao Huang. Least energy solutions for fractional Kirchhoff type equations involving critical growth. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1929-1954. doi: 10.3934/dcdss.2019126
[17]	Nguyen Huy Tuan. On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020354
[18]	Hongjun Gao, Šárka Nečasová, Tong Tang. On weak-strong uniqueness and singular limit for the compressible Primitive Equations. Discrete & Continuous Dynamical Systems, 2020, 40 (7) : 4287-4305. doi: 10.3934/dcds.2020181
[19]	Honglv Ma, Jin Zhang, Chengkui Zhong. Global existence and asymptotic behavior of global smooth solutions to the Kirchhoff equations with strong nonlinear damping. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4721-4737. doi: 10.3934/dcdsb.2019027
[20]	Andrea Signori. Penalisation of long treatment time and optimal control of a tumour growth model of Cahn–Hilliard type with singular potential. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2519-2542. doi: 10.3934/dcds.2020373

American Institute of Mathematical Sciences

Kirchhoff type equations with strong singularities

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Kirchhoff type equations with strong singularities

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors