Kirchhoff type equations with strong singularities

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

References:

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[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.
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[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.
[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.
[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.
[7]	J. Chabrowski, On the Neumann problem with singular and superlinear nonlinearities, Comm. in Applied Analysis, 13 (2009), 327-340.
[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.
[9]	M. Coclite and G. Palmieri, On a singular nonlinear Dirichlet problem, Comm. Partial Differential Equations, 14 (1989), 1315-1327. doi: 10.1080/03605308908820656.
[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.
[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.
[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.
[13]	I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353. doi: 10.1016/0022-247X(74)90025-0.
[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.
[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.
[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.
[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.
[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.
[19]	J. Giacomoni and K. Sreenadh, Multiplicity results for a singular and quasilinear equation, Discrete Continuous Dynamical Systems, (2007), 429-435.
[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.
[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.
[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.
[23]	D. Hai, On an asymptotically linear singular boundary value problems, Topological Methods in Nonlinear Analysis, 39 (2012), 83-92.
[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.
[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)
[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.
[27]	G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[37]	L. Xing and S. Yijing, Multiple positive solutions for Kirchhoff type problems with singularity, Comm. Pure Appl. Anal., 12 (2013), 721-733.
[38]	S. Yijing, Compatibility phenomena in singular problems, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 1321-1330. doi: 10.1017/S030821051100117X.
[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.
[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.
[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.
[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.
[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.

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.
[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.
[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.
[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.
[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.
[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.
[7]	J. Chabrowski, On the Neumann problem with singular and superlinear nonlinearities, Comm. in Applied Analysis, 13 (2009), 327-340.
[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.
[9]	M. Coclite and G. Palmieri, On a singular nonlinear Dirichlet problem, Comm. Partial Differential Equations, 14 (1989), 1315-1327. doi: 10.1080/03605308908820656.
[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.
[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.
[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.
[13]	I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353. doi: 10.1016/0022-247X(74)90025-0.
[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.
[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.
[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.
[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.
[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.
[19]	J. Giacomoni and K. Sreenadh, Multiplicity results for a singular and quasilinear equation, Discrete Continuous Dynamical Systems, (2007), 429-435.
[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.
[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.
[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.
[23]	D. Hai, On an asymptotically linear singular boundary value problems, Topological Methods in Nonlinear Analysis, 39 (2012), 83-92.
[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.
[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)
[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.
[27]	G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[37]	L. Xing and S. Yijing, Multiple positive solutions for Kirchhoff type problems with singularity, Comm. Pure Appl. Anal., 12 (2013), 721-733.
[38]	S. Yijing, Compatibility phenomena in singular problems, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 1321-1330. doi: 10.1017/S030821051100117X.
[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.
[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.
[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.
[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.
[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.

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