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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 - A, 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. 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.

[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. 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. 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. 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.

[8]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state,, Arch. Rational Mech. Anal., 82 (1983), 313. doi: 10.1007/BF00250555.

[9]

J. Byeon, Singularly perturbed nonlinear Dirichlet problems with a general nonlinearity,, Trans. Amer. Math. Soc., 362 (2010), 1981. 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. 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. 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. 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, (1998).

[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.

[15]

G. Kirchhoff, Mechanik,, Teubner, (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. 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.

[18]

J. Jin and X. Wu, Infinitely many radial solutions for Kirchhoff-type problems in $\mathbbR^N$,, J. Math. Anal. Appl., 369 (2010), 564. 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. 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. 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., 30 (1978), 284.

[22]

W. Liu and X. He, Multiplicity of high energy solutions for superlinear Kirchhoff equations,, J. Appl. Math. Comput., 39 (2012), 473. doi: 10.1007/s12190-012-0536-1.

[23]

T. F. Ma, Remarks on an elliptic equation of Kirchhoff type,, Nonlinear Anal., 63 (2005). 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. 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 (1975), 152.

[26]

D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term,, J. Funct. Anal., 237 (2006), 655. doi: 10.1016/j.jfa.2006.04.005.

[27]

M. Willem, Minimax Theorems,, Progress in Nonlinear Differential Equations and Their Applications, 24 (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. doi: 10.1016/j.nonrwa.2010.09.023.

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. 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.

[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. 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. 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. 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.

[8]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state,, Arch. Rational Mech. Anal., 82 (1983), 313. doi: 10.1007/BF00250555.

[9]

J. Byeon, Singularly perturbed nonlinear Dirichlet problems with a general nonlinearity,, Trans. Amer. Math. Soc., 362 (2010), 1981. 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. 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. 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. 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, (1998).

[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.

[15]

G. Kirchhoff, Mechanik,, Teubner, (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. 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.

[18]

J. Jin and X. Wu, Infinitely many radial solutions for Kirchhoff-type problems in $\mathbbR^N$,, J. Math. Anal. Appl., 369 (2010), 564. 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. 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. 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., 30 (1978), 284.

[22]

W. Liu and X. He, Multiplicity of high energy solutions for superlinear Kirchhoff equations,, J. Appl. Math. Comput., 39 (2012), 473. doi: 10.1007/s12190-012-0536-1.

[23]

T. F. Ma, Remarks on an elliptic equation of Kirchhoff type,, Nonlinear Anal., 63 (2005). 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. 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 (1975), 152.

[26]

D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term,, J. Funct. Anal., 237 (2006), 655. doi: 10.1016/j.jfa.2006.04.005.

[27]

M. Willem, Minimax Theorems,, Progress in Nonlinear Differential Equations and Their Applications, 24 (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. doi: 10.1016/j.nonrwa.2010.09.023.

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