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March  2015, 35(3): 943-966. doi: 10.3934/dcds.2015.35.943

## Existence of ground state solutions to the nonlinear Kirchhoff type equations with potentials

 1 Mathematical Institute, Tohoku University, 6-3, Aoba, Aramaki, Aoba-ku, Sendai-Shi, Miyagi 980-8578, Japan

Received  February 2014 Revised  May 2014 Published  October 2014

In this paper, we study the existence of ground state solutions to the nonlinear Kirchhoff type equations $- m \left( \| \nabla u \|_{L^2(\mathbf{R}^N)}^{2} \right) \Delta u + V(x) u = |u|^{p-1} u \quad {\rm in}\ \mathbf{R}^N, \ u \in H^1(\mathbf{R}^N), \ N \geq 1$ where $1 < p < \infty$ when $N=1,2$, $1 < p < (N+2)/(N-2)$ when $N \geq 3$, $m: [0,\infty) \to (0,\infty)$ is a continuous function and $V:\mathbf{R}^N \to \mathbf{R}$ a smooth function. Under suitable conditions on $m(s)$ and $V$, it is shown that a ground state solution to the above equation exists.
Citation: Norihisa Ikoma. Existence of ground state solutions to the nonlinear Kirchhoff type equations with potentials. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 943-966. doi: 10.3934/dcds.2015.35.943
##### References:
 [1] C. O. Alves, F. J. S. A. Corrêa and T. F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl., 49 (2005), 85-93. doi: 10.1016/j.camwa.2005.01.008. [2] A. Azzollini, The elliptic Kirchhoff equation in $\mathbbR^N$ perturbed by a local nonlinearity, Differential and Integral Equations, 25 (2012), 543-554. [3] A. Azzollini, A note on the elliptic Kirchhoff equation in $\mathbbR^N$ perturbed by a local nonlinearity,, to appear in Commun. Contemp. Math., (). [4] A. Azzollini, P. d'Avenia and A. Pomponio, Multiple critical points for a class of nonlinear functionals, Ann. Mat. Pura Appl. (4), 190 (2011), 507-523. doi: 10.1007/s10231-010-0160-3. [5] A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 345 (2008), 90-108. doi: 10.1016/j.jmaa.2008.03.057. [6] T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbfR^N$, Comm. Partial Differential Equations, 20 (1995), 1725-1741. doi: 10.1080/03605309508821149. [7] H. Berestycki, T. Gallouët and O. Kavian, Équations de champs scalaires euclidiens non linéaires dans le plan (French) [Nonlinear Euclidean scalar field equations in the plane], C. R. Acad. Sci. Paris Sér. I Math., 297 (1983), 307-310. [8] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555. [9] J. Byeon, Singularly perturbed nonlinear Dirichlet problems with a general nonlinearity, Trans. Amer. Math. Soc., 362 (2010), 1981-2001. doi: 10.1090/S0002-9947-09-04746-1. [10] J. Byeon, L. Jeanjean and K. Tanaka, Standing waves for nonlinear Schrödinger equations with a general nonlinearity: one and two dimensional cases, Comm. Partial Differential Equations, 33 (2008), 1113-1136. doi: 10.1080/03605300701518174. [11] P. Caldiroli, A new proof of the existence of homoclinic orbits for a class of autonomous second order Hamiltonian systems in $\mathbfR^N$, Math. Nachr., 187 (1997), 19-27. doi: 10.1002/mana.19971870103. [12] G. M. Figueiredo, N. Ikoma and J. R. Santos Júnior, Existence and concentration result for the Kirchhoff type equations with general nonlinearities, Arch. Rational Mech. Anal., 213 (2014), 931-979. doi: 10.1007/s00205-014-0747-8. [13] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001. [14] J. Hirata, N. Ikoma and K. Tanaka, Nonlinear scalar field equations in $\mathbbR^N$: Mountain pass and symmetric mountain pass approaches, Topol. Methods Nonlinear Anal., 35 (2010), 253-276. [15] G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. [16] L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on $\mathbfR^N$, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 787-809. doi: 10.1017/S0308210500013147. [17] L. Jeanjean and K. Tanaka, A note on a mountain pass characterization of least energy solutions, Adv. Nonlinear Stud., 3 (2003), 445-455. [18] J. Jin and X. Wu, Infinitely many radial solutions for Kirchhoff-type problems in $\mathbbR^N$, J. Math. Anal. Appl., 369 (2010), 564-574. doi: 10.1016/j.jmaa.2010.03.059. [19] G. Li and H. Ye, Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $\mathbbR^3$, J. Differential Equations, 257 (2014), 566-600. doi: 10.1016/j.jde.2014.04.011. [20] Y. Li, F. Li and J. Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differential Equations, 253 (2012), 2285-2294. doi: 10.1016/j.jde.2012.05.017. [21] J.-L. Lions, On some questions in boundary value problems of mathematical physics, Contemporary developments in continuum mechanics and partial differential equations (Proc. Internat. Sympos., Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977), North-Holland Math. Stud., 30, North-Holland, Amsterdam-New York, (1978), 284-346, [22] W. Liu and X. He, Multiplicity of high energy solutions for superlinear Kirchhoff equations, J. Appl. Math. Comput., 39 (2012), 473-487. doi: 10.1007/s12190-012-0536-1. [23] T. F. Ma, Remarks on an elliptic equation of Kirchhoff type, Nonlinear Anal., 63 (2005), e1967-e1977. doi: 10.1016/j.na.2005.03.021. [24] 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. [25] S. I. Pohozaev, A certain class of quasilinear hyperbolic equations (Russian), Mat. Sb. (N.S.), 96 (138) (1975), 152-166, 168. [26] D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237 (2006), 655-674. doi: 10.1016/j.jfa.2006.04.005. [27] M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications, Vol. 24, Birkhäuser Boston Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1. [28] X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in $\mathbfR^N$, Nonlinear Anal. Real World Appl., 12 (2011), 1278-1287. doi: 10.1016/j.nonrwa.2010.09.023.

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##### References:
 [1] C. O. Alves, F. J. S. A. Corrêa and T. F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl., 49 (2005), 85-93. doi: 10.1016/j.camwa.2005.01.008. [2] A. Azzollini, The elliptic Kirchhoff equation in $\mathbbR^N$ perturbed by a local nonlinearity, Differential and Integral Equations, 25 (2012), 543-554. [3] A. Azzollini, A note on the elliptic Kirchhoff equation in $\mathbbR^N$ perturbed by a local nonlinearity,, to appear in Commun. Contemp. Math., (). [4] A. Azzollini, P. d'Avenia and A. Pomponio, Multiple critical points for a class of nonlinear functionals, Ann. Mat. Pura Appl. (4), 190 (2011), 507-523. doi: 10.1007/s10231-010-0160-3. [5] A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 345 (2008), 90-108. doi: 10.1016/j.jmaa.2008.03.057. [6] T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbfR^N$, Comm. Partial Differential Equations, 20 (1995), 1725-1741. doi: 10.1080/03605309508821149. [7] H. Berestycki, T. Gallouët and O. Kavian, Équations de champs scalaires euclidiens non linéaires dans le plan (French) [Nonlinear Euclidean scalar field equations in the plane], C. R. Acad. Sci. Paris Sér. I Math., 297 (1983), 307-310. [8] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555. [9] J. Byeon, Singularly perturbed nonlinear Dirichlet problems with a general nonlinearity, Trans. Amer. Math. Soc., 362 (2010), 1981-2001. doi: 10.1090/S0002-9947-09-04746-1. [10] J. Byeon, L. Jeanjean and K. Tanaka, Standing waves for nonlinear Schrödinger equations with a general nonlinearity: one and two dimensional cases, Comm. Partial Differential Equations, 33 (2008), 1113-1136. doi: 10.1080/03605300701518174. [11] P. Caldiroli, A new proof of the existence of homoclinic orbits for a class of autonomous second order Hamiltonian systems in $\mathbfR^N$, Math. Nachr., 187 (1997), 19-27. doi: 10.1002/mana.19971870103. [12] G. M. Figueiredo, N. Ikoma and J. R. Santos Júnior, Existence and concentration result for the Kirchhoff type equations with general nonlinearities, Arch. Rational Mech. Anal., 213 (2014), 931-979. doi: 10.1007/s00205-014-0747-8. [13] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001. [14] J. Hirata, N. Ikoma and K. Tanaka, Nonlinear scalar field equations in $\mathbbR^N$: Mountain pass and symmetric mountain pass approaches, Topol. Methods Nonlinear Anal., 35 (2010), 253-276. [15] G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. [16] L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on $\mathbfR^N$, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 787-809. doi: 10.1017/S0308210500013147. [17] L. Jeanjean and K. Tanaka, A note on a mountain pass characterization of least energy solutions, Adv. Nonlinear Stud., 3 (2003), 445-455. [18] J. Jin and X. Wu, Infinitely many radial solutions for Kirchhoff-type problems in $\mathbbR^N$, J. Math. Anal. Appl., 369 (2010), 564-574. doi: 10.1016/j.jmaa.2010.03.059. [19] G. Li and H. Ye, Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $\mathbbR^3$, J. Differential Equations, 257 (2014), 566-600. doi: 10.1016/j.jde.2014.04.011. [20] Y. Li, F. Li and J. Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differential Equations, 253 (2012), 2285-2294. doi: 10.1016/j.jde.2012.05.017. [21] J.-L. Lions, On some questions in boundary value problems of mathematical physics, Contemporary developments in continuum mechanics and partial differential equations (Proc. Internat. Sympos., Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977), North-Holland Math. Stud., 30, North-Holland, Amsterdam-New York, (1978), 284-346, [22] W. Liu and X. He, Multiplicity of high energy solutions for superlinear Kirchhoff equations, J. Appl. Math. Comput., 39 (2012), 473-487. doi: 10.1007/s12190-012-0536-1. [23] T. F. Ma, Remarks on an elliptic equation of Kirchhoff type, Nonlinear Anal., 63 (2005), e1967-e1977. doi: 10.1016/j.na.2005.03.021. [24] 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. [25] S. I. Pohozaev, A certain class of quasilinear hyperbolic equations (Russian), Mat. Sb. (N.S.), 96 (138) (1975), 152-166, 168. [26] D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237 (2006), 655-674. doi: 10.1016/j.jfa.2006.04.005. [27] M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications, Vol. 24, Birkhäuser Boston Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1. [28] X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in $\mathbfR^N$, Nonlinear Anal. Real World Appl., 12 (2011), 1278-1287. doi: 10.1016/j.nonrwa.2010.09.023.
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