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Kirchhoff type equations with strong singularities
Department of Mathematics, University of Chinese Academy of Sciences, Beijing 100049, China |
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)$.
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) 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. |
[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. |
[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) 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. |
[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. |
[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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