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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 $\mathbb{R}^2$

 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

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.
Citation: Sami Aouaoui. A multiplicity result for some Kirchhoff-type equations involving exponential growth condition in $\mathbb{R}^2$. Communications on Pure & 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,, \emph{Ann. Scuola. Norm. Sup. Pisa}, 17 (1990), 481.   Google Scholar [2] C.O. Alves, Multiplicity of solutions for a class of elliptic problem in $\mathbbR^2$ with Neumann conditions,, \emph{J. Differential Equations}, 219 (2005), 20.  doi: 10.1016/j.jde.2004.11.010.  Google Scholar [3] C.O. Alves and D.S. Pereira, Existence and nonexistence of least energy nodal solution for a class of elliptic problem in $\mathbbR^2,$, \emph{Topol. Methods Nonlinear Anal.}, 46 (2015), 867.   Google Scholar [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), (2014).   Google Scholar [5] C.O. Alves and S.H.M. Soares, Nodal solutions for singularly perturbed equations with critical exponential growth., \emph{J. Differential Equation}, 234 (2007), 464.   Google Scholar [6] S. Aouaoui, Existence of multiple solutions to elliptic problems of Kirchhoff type with critical exponential growth,, \emph{Electon. J. Differential Equations}, 2014 (2014), 1.   Google Scholar [7] S. Aouaoui, On some nonlocal problem involving the N-Laplacian in $\mathbbR^N,$, \emph{Nonlinear Stud.}, 22 (2015), 57.   Google Scholar [8] S. Aouaoui, A multiplicity result for some nonlocal eigenvalue problem with exponential growth condition,, \emph{Nonlinear Anal.}, 125 (2015), 626.   Google Scholar [9] T. Bartsch, T. Weth and M. Willem, Partial symmetry of least energy nodal solutions to some variational problems,, \emph{J. Anal. Math.}, 96 (2005), 1.   Google Scholar [10] T. Bartsch and T. Weth, Three nodal solutions of singularly perturbed elliptic equations on domains without topology,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 22 (2005), 259.  doi: 10.1016/j.anihpc.2004.07.005.  Google Scholar [11] T. Bartsch and Z.-Q. Wang, Sign changing solutions of nonlinear Schrödinger equations,, \emph{Topol. Methods Nonlinear Anal.}, 13 (1999), 191.   Google Scholar [12] T. Bartsch and T. Weth, A note on additional properties of sign changing solutions to superlinear elliptic equations,, \emph{Topol. Methods Nonlinear Anal.}, 22 (2003), 1.   Google Scholar [13] C.J. Batkam, Multiple sign-changing solutions to a class of Kirchhoff type problems,, (2015) arXiv:1501.05733v1., (2015).   Google Scholar [14] C.J. Batkam, An elliptic equation under the effect of two nonlocal terms,, \emph{Math. Meth. Appl. Sci.}, (2015).   Google Scholar [15] H. Beresticky and P.L. Lions, Nonlinear scalar field equations, I. Existence of a ground state,, \emph{Arch. Ration. Mech. Anal.}, 82 (1983), 313.   Google Scholar [16] H. Brezis, Analyse Fonctionnelle (théorie et applications),, Masson, (1983).   Google Scholar [17] D.M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in $\mathbbR^2$,, \emph{Commun. Partial Differ. Equ.}, 17 (1992), 407.   Google Scholar [18] E.N. Dancer and Z. Zhitao, Fucik spectrum, sign-changing, and multiple solutions for semilinear elliptic boundary value problems with resonance at infinity,, \emph{J. Math. Anal. Appl.}, 250 (2000), 449.  doi: 10.1006/jmaa.2000.6969.  Google Scholar [19] J.M. do Ó, $N$-Laplacian equations in $\mathbbR^N$ with critical growth,, \emph{Abstr. Appl. Anal.}, 2 (1997), 301.  doi: 10.1155/S1085337597000419.  Google Scholar [20] J.M. do Ó, E. Medeiros and U. Severo, A nonhomogeneous elliptic problem involving critical growth in dimension two,, \emph{J. Math. Anal. Appl.}, 345 (2008), 286.  doi: 10.1016/j.jmaa.2008.03.074.  Google Scholar [21] G.M. Figueiredo and R.G. Nascimento, Existence of a nodal solution with minimal energy for a Kirchhoff equation,, \emph{Math. Nachr.}, 288 (2015), 48.   Google Scholar [22] G.M. Figueiredo and U.B. Severo, Ground state solution for a Kirchhoff problem with exponential critical growth,, \emph{Milan J. Math.}, (2015).  doi: 10.10007/s00032-015-0248-8.  Google Scholar [23] M.E. Filippakis and N.S. Papageorgiou, Multiple constant sign and nodal solutions for nonlinear elliptic equations with the p-Laplacian,, \emph{J. Differential Equations}, 245 (2008), 1883.  doi: 10.1016/j.jde.2008.07.004.  Google Scholar [24] Q. Li and Z. Yang, Multiple solutions for N-Kirchhoff type problems with critical exponential growth in $\mathbbR^N$,, \emph{Nonlinear Anal.}, 117 (2015), 159.  doi: 10.1016/j.na.2015.01.005.  Google Scholar [25] X. Li and X. He, Multiple sign-changing solutions for Kirchhoff-type equations,, \emph{Discrete Dyn. Nat. Soc.}, (2015), 1.   Google Scholar [26] Z. Liu and J. Sun, Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations,, \emph{J. Differential Equations}, 172 (2001), 257.   Google Scholar [27] A. Mao and S. Yuan, Sign-changing solutions of a class of nonlocal quasilinear elliptic boundary value problems,, \emph{J. Math. Anal. Appl.}, 383 (2011), 239.  doi: 10.1016/j.jmaa.2011.05.021.  Google Scholar [28] D. Mugnai, Four nontrivial solutions for subcritical exponential equation,, \emph{Calc. Var. Partial Differential Equations}, 32 (2008), 480.  doi: 10.1007/s00526-007-0148-z.  Google Scholar [29] R. Pei and J. Zhang, Nontrivial solutions for asymmetric Kirchhoff type problems,, \emph{Abstr. Appl. Anal}, 2014 (2014).  doi: 10.1155/2014/163645.  Google Scholar [30] K. Sreenadh and S. Goyal, $n$-Kirchhoff type equations with exponential nonlinearities,, \emph{Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM}, (2015), 13398.   Google Scholar [31] S. Struwe, Variational Methods,, Springer-Verlag, (2000).  doi: 10.1007/978-3-662-04194-9.  Google Scholar [32] N.S. Trudinger, On Harnack type inequalities and their applications to quasilinear elliptic equations,, \emph{Comm. Pure Appl. Math.}, XX (1967), 721.   Google Scholar [33] T. Weth, Nodal solutions to superlinear biharmonic equations via decomposition in dual cones,, \emph{Topol. Methods Nonlinear Anal.}, 28 (2006), 33.   Google Scholar [34] Y. Wu and Y. Huang, Sign-changing solutions for Schroinger equations with indefinite superlinear nonlinearities,, \emph{J. Math. Anal. Appl.}, 401 (2013), 850.  doi: 10.1016/j.jmaa.2013.01.006.  Google Scholar [35] W. Zhang and X. Liu, Infinitely many sign-changing solutions for a quasilinear elliptic equation in $\mathbbR^N$,, \emph{J. Math. Anal. Appl.}, 427 (2015), 722.  doi: 10.1016/j.jmaa.2015.02.070.  Google Scholar [36] Z. Zhang, M. Calanchi and B. Ruf, Elliptic equations in $\mathbbR^2$ with one-sided exponential growth,, \emph{Commun. Contemp. Math.}, 6 (2004), 947.  doi: 10.1142/S0219199704001549.  Google Scholar [37] Z. Zhang and K. Perera, Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow,, \emph{J. Math. Anal. Appl.}, 317 (2006), 456.  doi: 10.1016/j.jmaa.2005.06.102.  Google Scholar

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,, \emph{Ann. Scuola. Norm. Sup. Pisa}, 17 (1990), 481.   Google Scholar [2] C.O. Alves, Multiplicity of solutions for a class of elliptic problem in $\mathbbR^2$ with Neumann conditions,, \emph{J. Differential Equations}, 219 (2005), 20.  doi: 10.1016/j.jde.2004.11.010.  Google Scholar [3] C.O. Alves and D.S. Pereira, Existence and nonexistence of least energy nodal solution for a class of elliptic problem in $\mathbbR^2,$, \emph{Topol. Methods Nonlinear Anal.}, 46 (2015), 867.   Google Scholar [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), (2014).   Google Scholar [5] C.O. Alves and S.H.M. Soares, Nodal solutions for singularly perturbed equations with critical exponential growth., \emph{J. Differential Equation}, 234 (2007), 464.   Google Scholar [6] S. Aouaoui, Existence of multiple solutions to elliptic problems of Kirchhoff type with critical exponential growth,, \emph{Electon. J. Differential Equations}, 2014 (2014), 1.   Google Scholar [7] S. Aouaoui, On some nonlocal problem involving the N-Laplacian in $\mathbbR^N,$, \emph{Nonlinear Stud.}, 22 (2015), 57.   Google Scholar [8] S. Aouaoui, A multiplicity result for some nonlocal eigenvalue problem with exponential growth condition,, \emph{Nonlinear Anal.}, 125 (2015), 626.   Google Scholar [9] T. Bartsch, T. Weth and M. Willem, Partial symmetry of least energy nodal solutions to some variational problems,, \emph{J. Anal. Math.}, 96 (2005), 1.   Google Scholar [10] T. Bartsch and T. Weth, Three nodal solutions of singularly perturbed elliptic equations on domains without topology,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 22 (2005), 259.  doi: 10.1016/j.anihpc.2004.07.005.  Google Scholar [11] T. Bartsch and Z.-Q. Wang, Sign changing solutions of nonlinear Schrödinger equations,, \emph{Topol. Methods Nonlinear Anal.}, 13 (1999), 191.   Google Scholar [12] T. Bartsch and T. Weth, A note on additional properties of sign changing solutions to superlinear elliptic equations,, \emph{Topol. Methods Nonlinear Anal.}, 22 (2003), 1.   Google Scholar [13] C.J. Batkam, Multiple sign-changing solutions to a class of Kirchhoff type problems,, (2015) arXiv:1501.05733v1., (2015).   Google Scholar [14] C.J. Batkam, An elliptic equation under the effect of two nonlocal terms,, \emph{Math. Meth. Appl. Sci.}, (2015).   Google Scholar [15] H. Beresticky and P.L. Lions, Nonlinear scalar field equations, I. Existence of a ground state,, \emph{Arch. Ration. Mech. Anal.}, 82 (1983), 313.   Google Scholar [16] H. Brezis, Analyse Fonctionnelle (théorie et applications),, Masson, (1983).   Google Scholar [17] D.M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in $\mathbbR^2$,, \emph{Commun. Partial Differ. Equ.}, 17 (1992), 407.   Google Scholar [18] E.N. Dancer and Z. Zhitao, Fucik spectrum, sign-changing, and multiple solutions for semilinear elliptic boundary value problems with resonance at infinity,, \emph{J. Math. Anal. Appl.}, 250 (2000), 449.  doi: 10.1006/jmaa.2000.6969.  Google Scholar [19] J.M. do Ó, $N$-Laplacian equations in $\mathbbR^N$ with critical growth,, \emph{Abstr. Appl. Anal.}, 2 (1997), 301.  doi: 10.1155/S1085337597000419.  Google Scholar [20] J.M. do Ó, E. Medeiros and U. Severo, A nonhomogeneous elliptic problem involving critical growth in dimension two,, \emph{J. Math. Anal. Appl.}, 345 (2008), 286.  doi: 10.1016/j.jmaa.2008.03.074.  Google Scholar [21] G.M. Figueiredo and R.G. Nascimento, Existence of a nodal solution with minimal energy for a Kirchhoff equation,, \emph{Math. Nachr.}, 288 (2015), 48.   Google Scholar [22] G.M. Figueiredo and U.B. Severo, Ground state solution for a Kirchhoff problem with exponential critical growth,, \emph{Milan J. Math.}, (2015).  doi: 10.10007/s00032-015-0248-8.  Google Scholar [23] M.E. Filippakis and N.S. Papageorgiou, Multiple constant sign and nodal solutions for nonlinear elliptic equations with the p-Laplacian,, \emph{J. Differential Equations}, 245 (2008), 1883.  doi: 10.1016/j.jde.2008.07.004.  Google Scholar [24] Q. Li and Z. Yang, Multiple solutions for N-Kirchhoff type problems with critical exponential growth in $\mathbbR^N$,, \emph{Nonlinear Anal.}, 117 (2015), 159.  doi: 10.1016/j.na.2015.01.005.  Google Scholar [25] X. Li and X. He, Multiple sign-changing solutions for Kirchhoff-type equations,, \emph{Discrete Dyn. Nat. Soc.}, (2015), 1.   Google Scholar [26] Z. Liu and J. Sun, Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations,, \emph{J. Differential Equations}, 172 (2001), 257.   Google Scholar [27] A. Mao and S. Yuan, Sign-changing solutions of a class of nonlocal quasilinear elliptic boundary value problems,, \emph{J. Math. Anal. Appl.}, 383 (2011), 239.  doi: 10.1016/j.jmaa.2011.05.021.  Google Scholar [28] D. Mugnai, Four nontrivial solutions for subcritical exponential equation,, \emph{Calc. Var. Partial Differential Equations}, 32 (2008), 480.  doi: 10.1007/s00526-007-0148-z.  Google Scholar [29] R. Pei and J. Zhang, Nontrivial solutions for asymmetric Kirchhoff type problems,, \emph{Abstr. Appl. Anal}, 2014 (2014).  doi: 10.1155/2014/163645.  Google Scholar [30] K. Sreenadh and S. Goyal, $n$-Kirchhoff type equations with exponential nonlinearities,, \emph{Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM}, (2015), 13398.   Google Scholar [31] S. Struwe, Variational Methods,, Springer-Verlag, (2000).  doi: 10.1007/978-3-662-04194-9.  Google Scholar [32] N.S. Trudinger, On Harnack type inequalities and their applications to quasilinear elliptic equations,, \emph{Comm. Pure Appl. Math.}, XX (1967), 721.   Google Scholar [33] T. Weth, Nodal solutions to superlinear biharmonic equations via decomposition in dual cones,, \emph{Topol. Methods Nonlinear Anal.}, 28 (2006), 33.   Google Scholar [34] Y. Wu and Y. Huang, Sign-changing solutions for Schroinger equations with indefinite superlinear nonlinearities,, \emph{J. Math. Anal. Appl.}, 401 (2013), 850.  doi: 10.1016/j.jmaa.2013.01.006.  Google Scholar [35] W. Zhang and X. Liu, Infinitely many sign-changing solutions for a quasilinear elliptic equation in $\mathbbR^N$,, \emph{J. Math. Anal. Appl.}, 427 (2015), 722.  doi: 10.1016/j.jmaa.2015.02.070.  Google Scholar [36] Z. Zhang, M. Calanchi and B. Ruf, Elliptic equations in $\mathbbR^2$ with one-sided exponential growth,, \emph{Commun. Contemp. Math.}, 6 (2004), 947.  doi: 10.1142/S0219199704001549.  Google Scholar [37] Z. Zhang and K. Perera, Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow,, \emph{J. Math. Anal. Appl.}, 317 (2006), 456.  doi: 10.1016/j.jmaa.2005.06.102.  Google Scholar
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