A multiplicity result for some Kirchhoff-type equations involving exponential growth condition in \mathbb{R}^2

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July 2016, 15(4): 1351-1370. doi: 10.3934/cpaa.2016.15.1351

A multiplicity result for some Kirchhoff-type equations involving exponential growth condition in $\mathbb{R}^2 $

Sami Aouaoui ^1,

1.	Institut Supérieur des Mathématiques Appliquées et de l'Informatique de Kairouan, Avenue Assad Iben Fourat, 3100 Kairouan, Tunisia

Received November 2015 Revised January 2016 Published April 2016

Abstract
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In this paper, we consider the existence and multiplicity of sign-changing solutions to some Kirchhoff-type equation involving a nonlinear term with exponential growth. In a first result, we prove the existence of at least three solutions: one solution is positive, one is negative and the third one is sign-changing. The existence of infinitely many sign-changing solutions is proved as our second result in this work. Our method is mainly based on invariant sets of descending flow in the framework of classical critical point theory.

Keywords: multiplicity, Kirchhoff-type, sign-changing solution, invariant sets, descending flow, critical points., Trudinger-Moser inequality.

Mathematics Subject Classification: Primary: 35J20, 35J62; Secondary: 35A15, 35B08, 35D3.

Citation: Sami Aouaoui. A multiplicity result for some Kirchhoff-type equations involving exponential growth condition in $\mathbb{R}^2 $. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1351-1370. doi: 10.3934/cpaa.2016.15.1351

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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