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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 & 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.  Google Scholar

[2]

A. Azzollini, The elliptic Kirchhoff equation in $\mathbbR^N$ perturbed by a local nonlinearity, Differential and Integral Equations, 25 (2012), 543-554.  Google Scholar

[3]

A. Azzollini, A note on the elliptic Kirchhoff equation in $\mathbbR^N$ perturbed by a local nonlinearity,, to appear in Commun. Contemp. Math., ().   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001.  Google Scholar

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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.  Google Scholar

[15]

G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. Google Scholar

[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.  Google Scholar

[17]

L. Jeanjean and K. Tanaka, A note on a mountain pass characterization of least energy solutions, Adv. Nonlinear Stud., 3 (2003), 445-455.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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,  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

S. I. Pohozaev, A certain class of quasilinear hyperbolic equations (Russian), Mat. Sb. (N.S.), 96 (138) (1975), 152-166, 168.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[2]

A. Azzollini, The elliptic Kirchhoff equation in $\mathbbR^N$ perturbed by a local nonlinearity, Differential and Integral Equations, 25 (2012), 543-554.  Google Scholar

[3]

A. Azzollini, A note on the elliptic Kirchhoff equation in $\mathbbR^N$ perturbed by a local nonlinearity,, to appear in Commun. Contemp. Math., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. Google Scholar

[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.  Google Scholar

[17]

L. Jeanjean and K. Tanaka, A note on a mountain pass characterization of least energy solutions, Adv. Nonlinear Stud., 3 (2003), 445-455.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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,  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

S. I. Pohozaev, A certain class of quasilinear hyperbolic equations (Russian), Mat. Sb. (N.S.), 96 (138) (1975), 152-166, 168.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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