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A multiplicity result for some Kirchhoff-type equations involving exponential growth condition in $\mathbb{R}^2 $
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References:
[1] |
Adimurthi and S.L. Yadava, Multiplicity results for semilinear elliptic equations in bounded domain of $\mathbbR^2$ involving critical exponent, Ann. Scuola. Norm. Sup. Pisa, 17 (1990), 481-504. |
[2] |
C.O. Alves, Multiplicity of solutions for a class of elliptic problem in $ \mathbbR^2 $ with Neumann conditions, J. Differential Equations, 219 (2005), 20-39.
doi: 10.1016/j.jde.2004.11.010. |
[3] |
C.O. Alves and D.S. Pereira, Existence and nonexistence of least energy nodal solution for a class of elliptic problem in $ \mathbbR^2, $ Topol. Methods Nonlinear Anal., 46 (2015), 867-892. |
[4] |
C.O. Alves and D.S. Pereira, Multiplicity of multi-bump type nodal solutions for a class of elliptic problems with exponential critical growth in $\mathbbR^2,$ (2014), arXiv:1412.4219v1. |
[5] |
C.O. Alves and S.H.M. Soares, Nodal solutions for singularly perturbed equations with critical exponential growth. J. Differential Equation, 234 (2007), 464-484. |
[6] |
S. Aouaoui, Existence of multiple solutions to elliptic problems of Kirchhoff type with critical exponential growth, Electon. J. Differential Equations, 2014 (2014), 1-12. |
[7] |
S. Aouaoui, On some nonlocal problem involving the N-Laplacian in $ \mathbbR^N, $ Nonlinear Stud., 22 (2015), 57-70. |
[8] |
S. Aouaoui, A multiplicity result for some nonlocal eigenvalue problem with exponential growth condition, Nonlinear Anal., 125 (2015), 626-638. |
[9] |
T. Bartsch, T. Weth and M. Willem, Partial symmetry of least energy nodal solutions to some variational problems, J. Anal. Math., 96 (2005), 1-18. |
[10] |
T. Bartsch and T. Weth, Three nodal solutions of singularly perturbed elliptic equations on domains without topology, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 259-281.
doi: 10.1016/j.anihpc.2004.07.005. |
[11] |
T. Bartsch and Z.-Q. Wang, Sign changing solutions of nonlinear Schrödinger equations, Topol. Methods Nonlinear Anal., 13 (1999), 191-198. |
[12] |
T. Bartsch and T. Weth, A note on additional properties of sign changing solutions to superlinear elliptic equations, Topol. Methods Nonlinear Anal., 22 (2003), 1-14. |
[13] |
C.J. Batkam, Multiple sign-changing solutions to a class of Kirchhoff type problems, (2015) arXiv:1501.05733v1. |
[14] |
C.J. Batkam, An elliptic equation under the effect of two nonlocal terms, Math. Meth. Appl. Sci., (2015), doi: 10.1002/mma.3587. |
[15] |
H. Beresticky and P.L. Lions, Nonlinear scalar field equations, I. Existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-346. |
[16] |
H. Brezis, Analyse Fonctionnelle (théorie et applications), Masson, Paris, 1983. |
[17] |
D.M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in $ \mathbbR^2 $, Commun. Partial Differ. Equ., 17 (1992), 407-435. |
[18] |
E.N. Dancer and Z. Zhitao, Fucik spectrum, sign-changing, and multiple solutions for semilinear elliptic boundary value problems with resonance at infinity, J. Math. Anal. Appl., 250 (2000), 449-464.
doi: 10.1006/jmaa.2000.6969. |
[19] |
J.M. do Ó, $N$-Laplacian equations in $\mathbbR^N$ with critical growth, Abstr. Appl. Anal., 2 (1997), 301-315.
doi: 10.1155/S1085337597000419. |
[20] |
J.M. do Ó, E. Medeiros and U. Severo, A nonhomogeneous elliptic problem involving critical growth in dimension two, J. Math. Anal. Appl., 345 (2008), 286-304.
doi: 10.1016/j.jmaa.2008.03.074. |
[21] |
G.M. Figueiredo and R.G. Nascimento, Existence of a nodal solution with minimal energy for a Kirchhoff equation, Math. Nachr., 288 (2015), 48-60. |
[22] |
G.M. Figueiredo and U.B. Severo, Ground state solution for a Kirchhoff problem with exponential critical growth, Milan J. Math., (2015).
doi: 10.10007/s00032-015-0248-8. |
[23] |
M.E. Filippakis and N.S. Papageorgiou, Multiple constant sign and nodal solutions for nonlinear elliptic equations with the p-Laplacian, J. Differential Equations, 245 (2008), 1883-1922.
doi: 10.1016/j.jde.2008.07.004. |
[24] |
Q. Li and Z. Yang, Multiple solutions for N-Kirchhoff type problems with critical exponential growth in $\mathbbR^N$, Nonlinear Anal., 117 (2015), 159-168.
doi: 10.1016/j.na.2015.01.005. |
[25] |
X. Li and X. He, Multiple sign-changing solutions for Kirchhoff-type equations, Discrete Dyn. Nat. Soc., 2015, Article ID 985986, 1-9. |
[26] |
Z. Liu and J. Sun, Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations, J. Differential Equations, 172 (2001), 257-299. |
[27] |
A. Mao and S. Yuan, Sign-changing solutions of a class of nonlocal quasilinear elliptic boundary value problems, J. Math. Anal. Appl., 383 (2011), 239-243.
doi: 10.1016/j.jmaa.2011.05.021. |
[28] |
D. Mugnai, Four nontrivial solutions for subcritical exponential equation, Calc. Var. Partial Differential Equations, 32 (2008), 480-497.
doi: 10.1007/s00526-007-0148-z. |
[29] |
R. Pei and J. Zhang, Nontrivial solutions for asymmetric Kirchhoff type problems, Abstr. Appl. Anal, 2014 (2014), Article ID 163645.
doi: 10.1155/2014/163645. |
[30] |
K. Sreenadh and S. Goyal, $n$-Kirchhoff type equations with exponential nonlinearities, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM, (2015), DOI 10.1007/s13398-015-0230-x. |
[31] |
S. Struwe, Variational Methods, Springer-Verlag, Berlin, Heidelberg, 2000.
doi: 10.1007/978-3-662-04194-9. |
[32] |
N.S. Trudinger, On Harnack type inequalities and their applications to quasilinear elliptic equations, Comm. Pure Appl. Math., XX (1967), 721-747. |
[33] |
T. Weth, Nodal solutions to superlinear biharmonic equations via decomposition in dual cones, Topol. Methods Nonlinear Anal., 28 (2006), 33-52. |
[34] |
Y. Wu and Y. Huang, Sign-changing solutions for Schroinger equations with indefinite superlinear nonlinearities, J. Math. Anal. Appl., 401 (2013), 850-860.
doi: 10.1016/j.jmaa.2013.01.006. |
[35] |
W. Zhang and X. Liu, Infinitely many sign-changing solutions for a quasilinear elliptic equation in $\mathbbR^N$, J. Math. Anal. Appl., 427 (2015), 722-740.
doi: 10.1016/j.jmaa.2015.02.070. |
[36] |
Z. Zhang, M. Calanchi and B. Ruf, Elliptic equations in $\mathbbR^2$ with one-sided exponential growth, Commun. Contemp. Math., 6 (2004), 947-971.
doi: 10.1142/S0219199704001549. |
[37] |
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] |
Adimurthi and S.L. Yadava, Multiplicity results for semilinear elliptic equations in bounded domain of $\mathbbR^2$ involving critical exponent, Ann. Scuola. Norm. Sup. Pisa, 17 (1990), 481-504. |
[2] |
C.O. Alves, Multiplicity of solutions for a class of elliptic problem in $ \mathbbR^2 $ with Neumann conditions, J. Differential Equations, 219 (2005), 20-39.
doi: 10.1016/j.jde.2004.11.010. |
[3] |
C.O. Alves and D.S. Pereira, Existence and nonexistence of least energy nodal solution for a class of elliptic problem in $ \mathbbR^2, $ Topol. Methods Nonlinear Anal., 46 (2015), 867-892. |
[4] |
C.O. Alves and D.S. Pereira, Multiplicity of multi-bump type nodal solutions for a class of elliptic problems with exponential critical growth in $\mathbbR^2,$ (2014), arXiv:1412.4219v1. |
[5] |
C.O. Alves and S.H.M. Soares, Nodal solutions for singularly perturbed equations with critical exponential growth. J. Differential Equation, 234 (2007), 464-484. |
[6] |
S. Aouaoui, Existence of multiple solutions to elliptic problems of Kirchhoff type with critical exponential growth, Electon. J. Differential Equations, 2014 (2014), 1-12. |
[7] |
S. Aouaoui, On some nonlocal problem involving the N-Laplacian in $ \mathbbR^N, $ Nonlinear Stud., 22 (2015), 57-70. |
[8] |
S. Aouaoui, A multiplicity result for some nonlocal eigenvalue problem with exponential growth condition, Nonlinear Anal., 125 (2015), 626-638. |
[9] |
T. Bartsch, T. Weth and M. Willem, Partial symmetry of least energy nodal solutions to some variational problems, J. Anal. Math., 96 (2005), 1-18. |
[10] |
T. Bartsch and T. Weth, Three nodal solutions of singularly perturbed elliptic equations on domains without topology, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 259-281.
doi: 10.1016/j.anihpc.2004.07.005. |
[11] |
T. Bartsch and Z.-Q. Wang, Sign changing solutions of nonlinear Schrödinger equations, Topol. Methods Nonlinear Anal., 13 (1999), 191-198. |
[12] |
T. Bartsch and T. Weth, A note on additional properties of sign changing solutions to superlinear elliptic equations, Topol. Methods Nonlinear Anal., 22 (2003), 1-14. |
[13] |
C.J. Batkam, Multiple sign-changing solutions to a class of Kirchhoff type problems, (2015) arXiv:1501.05733v1. |
[14] |
C.J. Batkam, An elliptic equation under the effect of two nonlocal terms, Math. Meth. Appl. Sci., (2015), doi: 10.1002/mma.3587. |
[15] |
H. Beresticky and P.L. Lions, Nonlinear scalar field equations, I. Existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-346. |
[16] |
H. Brezis, Analyse Fonctionnelle (théorie et applications), Masson, Paris, 1983. |
[17] |
D.M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in $ \mathbbR^2 $, Commun. Partial Differ. Equ., 17 (1992), 407-435. |
[18] |
E.N. Dancer and Z. Zhitao, Fucik spectrum, sign-changing, and multiple solutions for semilinear elliptic boundary value problems with resonance at infinity, J. Math. Anal. Appl., 250 (2000), 449-464.
doi: 10.1006/jmaa.2000.6969. |
[19] |
J.M. do Ó, $N$-Laplacian equations in $\mathbbR^N$ with critical growth, Abstr. Appl. Anal., 2 (1997), 301-315.
doi: 10.1155/S1085337597000419. |
[20] |
J.M. do Ó, E. Medeiros and U. Severo, A nonhomogeneous elliptic problem involving critical growth in dimension two, J. Math. Anal. Appl., 345 (2008), 286-304.
doi: 10.1016/j.jmaa.2008.03.074. |
[21] |
G.M. Figueiredo and R.G. Nascimento, Existence of a nodal solution with minimal energy for a Kirchhoff equation, Math. Nachr., 288 (2015), 48-60. |
[22] |
G.M. Figueiredo and U.B. Severo, Ground state solution for a Kirchhoff problem with exponential critical growth, Milan J. Math., (2015).
doi: 10.10007/s00032-015-0248-8. |
[23] |
M.E. Filippakis and N.S. Papageorgiou, Multiple constant sign and nodal solutions for nonlinear elliptic equations with the p-Laplacian, J. Differential Equations, 245 (2008), 1883-1922.
doi: 10.1016/j.jde.2008.07.004. |
[24] |
Q. Li and Z. Yang, Multiple solutions for N-Kirchhoff type problems with critical exponential growth in $\mathbbR^N$, Nonlinear Anal., 117 (2015), 159-168.
doi: 10.1016/j.na.2015.01.005. |
[25] |
X. Li and X. He, Multiple sign-changing solutions for Kirchhoff-type equations, Discrete Dyn. Nat. Soc., 2015, Article ID 985986, 1-9. |
[26] |
Z. Liu and J. Sun, Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations, J. Differential Equations, 172 (2001), 257-299. |
[27] |
A. Mao and S. Yuan, Sign-changing solutions of a class of nonlocal quasilinear elliptic boundary value problems, J. Math. Anal. Appl., 383 (2011), 239-243.
doi: 10.1016/j.jmaa.2011.05.021. |
[28] |
D. Mugnai, Four nontrivial solutions for subcritical exponential equation, Calc. Var. Partial Differential Equations, 32 (2008), 480-497.
doi: 10.1007/s00526-007-0148-z. |
[29] |
R. Pei and J. Zhang, Nontrivial solutions for asymmetric Kirchhoff type problems, Abstr. Appl. Anal, 2014 (2014), Article ID 163645.
doi: 10.1155/2014/163645. |
[30] |
K. Sreenadh and S. Goyal, $n$-Kirchhoff type equations with exponential nonlinearities, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM, (2015), DOI 10.1007/s13398-015-0230-x. |
[31] |
S. Struwe, Variational Methods, Springer-Verlag, Berlin, Heidelberg, 2000.
doi: 10.1007/978-3-662-04194-9. |
[32] |
N.S. Trudinger, On Harnack type inequalities and their applications to quasilinear elliptic equations, Comm. Pure Appl. Math., XX (1967), 721-747. |
[33] |
T. Weth, Nodal solutions to superlinear biharmonic equations via decomposition in dual cones, Topol. Methods Nonlinear Anal., 28 (2006), 33-52. |
[34] |
Y. Wu and Y. Huang, Sign-changing solutions for Schroinger equations with indefinite superlinear nonlinearities, J. Math. Anal. Appl., 401 (2013), 850-860.
doi: 10.1016/j.jmaa.2013.01.006. |
[35] |
W. Zhang and X. Liu, Infinitely many sign-changing solutions for a quasilinear elliptic equation in $\mathbbR^N$, J. Math. Anal. Appl., 427 (2015), 722-740.
doi: 10.1016/j.jmaa.2015.02.070. |
[36] |
Z. Zhang, M. Calanchi and B. Ruf, Elliptic equations in $\mathbbR^2$ with one-sided exponential growth, Commun. Contemp. Math., 6 (2004), 947-971.
doi: 10.1142/S0219199704001549. |
[37] |
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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