August  2015, 35(8): 3857-3877. doi: 10.3934/dcds.2015.35.3857

Positive high energy solution for Kirchhoff equation in $\mathbb{R}^{3}$ with superlinear nonlinearities via Nehari-Pohožaev manifold

1. 

College of Science, Wuhan University of Science and Technology, Wuhan 430065, China

Received  June 2014 Revised  December 2014 Published  February 2015

In this paper, we study the following nonlinear problem of Kirchhoff type: \begin{equation}\label{(0.1)} \left\{% \begin{array}{ll} -\left(a+b\int\limits_{\mathbb{R}^3}|\nabla u|^2\right)\Delta u+V(x)u=f(u), & \hbox{$x\in \mathbb{R}^3$}, \\ u>0, & \hbox{$x\in \mathbb{R}^3$},                                 (0.1) \\ \end{array}% \right.\end{equation} where $a,$ $b>0$ are constants, $V:\mathbb{R}^3\rightarrow\mathbb{R}$ and $f(t)$ is subcritical and superlinear at infinity. Under certain assumptions on non-constant potential $V$, we prove the existence of positive high energy solutions by using a linking argument with a barycenter map restricted on a Nehari-Pohožaev type manifold.
    Our main result has solved Kirchhoff equation (0.1) with superlinear nonlinearities, which has not been studied, and can be viewed as a partial extension of a recent result of He and Zou in [9] concerning Kirchhoff equations with 4-superlinear nonlinearities.
Citation: Hongyu Ye. Positive high energy solution for Kirchhoff equation in $\mathbb{R}^{3}$ with superlinear nonlinearities via Nehari-Pohožaev manifold. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3857-3877. doi: 10.3934/dcds.2015.35.3857
References:
[1]

C. O. Alves and F. J. S. A. Correa, On existence of solutions for a class of problem involving a nonlinear operator,, Comm. Appl. Nonlinear Anal., 8 (2001), 43.   Google Scholar

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P. D'Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data,, Invent. Math., 108 (1992), 247.  doi: 10.1007/BF02100605.  Google Scholar

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A. Arosio and S. Panizzi, On the well-posedness of the Kirchhoff string,, Trans. Amer. Math. Soc., 348 (1996), 305.  doi: 10.1090/S0002-9947-96-01532-2.  Google Scholar

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S. Bernstein, Sur une classe d'équations fonctionelles aux dérivées partielles,, Bull. Acad. Sci. URSS. Sér., 4 (1940), 17.   Google Scholar

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M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. A. Soriano, Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation,, Adv. Differential Equations, 6 (2001), 701.   Google Scholar

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X. M. He and W. M. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in $\mathbbR^3$,, J. Differential Equations, 252 (2012), 1813.  doi: 10.1016/j.jde.2011.08.035.  Google Scholar

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L. Jeanjean, Existence of solutions with prescribed norm for semilinear elliptic equations,, Nonlinear Anal. Theory T. M. & A., 28 (1997), 1633.  doi: 10.1016/S0362-546X(96)00021-1.  Google Scholar

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J. H. 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.  Google Scholar

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

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Y. Li and W. M. Ni, Radial symmetry of positive solutions of nonlinear ellitic equations in $\mathbbR^n$,, Comm. Partial Differential Equations, 18 (1993), 1043.  doi: 10.1080/03605309308820960.  Google Scholar

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G. B. Li, Some properties of weak solutions of nonlinear scalar fields equation,, Ann. Acad. Sci. Fenn. Math., 15 (1990), 27.  doi: 10.5186/aasfm.1990.1521.  Google Scholar

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G. B. Li and H. Y. Ye, Existence of positive solutions for nonlinear Kirchhoff type problems in $\mathbbR^3$ with critical Sobolev exponent,, Math. Methods Appl. Sci., 37 (2014), 2570.  doi: 10.1002/mma.3000.  Google Scholar

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G. B. Li and H. Y. 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.  Google Scholar

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Y. H. Li et al., 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.  Google Scholar

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J. L. Lions, On some questions in boundary value problems of mathmatical physics,, in Contemporary Development in Continuum Mechanics and Partial Differential Equations, 30 (1978), 284.   Google Scholar

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

[20]

J. Moser, A New proof of de Giorgi's theorem concerning the regularity problem for elliptic differential equations,, Comm. Pure Appl. Math., 13 (1960), 457.  doi: 10.1002/cpa.3160130308.  Google Scholar

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S. I. Pohožaev, A certain class of quasilinear hyperbolic equations,, Mat. Sb. (NS), 96 (1975), 152.   Google Scholar

[22]

P. Tolksdorf, Regularity for some general class of quasilinear elliptic equations,, J. Differential Equations, 51 (1984), 126.  doi: 10.1016/0022-0396(84)90105-0.  Google Scholar

[23]

N. S. Trudinger, On Harnack type inequalities and their application to quasilinear ellitpic equations,, Comm. Pure Appl. Math., 20 (1967), 721.  doi: 10.1002/cpa.3160200406.  Google Scholar

[24]

J. Wang et al., Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth,, J. Differential Equations, 253 (2012), 2314.  doi: 10.1016/j.jde.2012.05.023.  Google Scholar

[25]

M. Willem, Minimax Theorems,, Progress in Nonlinear Differential Equations and their Applications, (1996).  doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[26]

X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in $\mathbbR^N$,, Nonlinear Anal.: Real World Applications, 12 (2011), 1278.  doi: 10.1016/j.nonrwa.2010.09.023.  Google Scholar

show all references

References:
[1]

C. O. Alves and F. J. S. A. Correa, On existence of solutions for a class of problem involving a nonlinear operator,, Comm. Appl. Nonlinear Anal., 8 (2001), 43.   Google Scholar

[2]

P. D'Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data,, Invent. Math., 108 (1992), 247.  doi: 10.1007/BF02100605.  Google Scholar

[3]

A. Arosio and S. Panizzi, On the well-posedness of the Kirchhoff string,, Trans. Amer. Math. Soc., 348 (1996), 305.  doi: 10.1090/S0002-9947-96-01532-2.  Google Scholar

[4]

P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity,, Nonlinear Anal., 7 (1983), 981.  doi: 10.1016/0362-546X(83)90115-3.  Google Scholar

[5]

T. Bartsch and T. Weth, Three nodal solutions of singularly perturbed elliptic equations on domains with topology,, Ann. I. H. Poincaré-AN, 22 (2005), 259.  doi: 10.1016/j.anihpc.2004.07.005.  Google Scholar

[6]

E. D. Benedetto, $C^{1+\alpha}$ local regularity of weak solutions of degenerate results for elliptic equations,, Nonlinear Anal., 7 (1983), 827.  doi: 10.1016/0362-546X(83)90061-5.  Google Scholar

[7]

S. Bernstein, Sur une classe d'équations fonctionelles aux dérivées partielles,, Bull. Acad. Sci. URSS. Sér., 4 (1940), 17.   Google Scholar

[8]

M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. A. Soriano, Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation,, Adv. Differential Equations, 6 (2001), 701.   Google Scholar

[9]

X. M. He and W. M. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in $\mathbbR^3$,, J. Differential Equations, 252 (2012), 1813.  doi: 10.1016/j.jde.2011.08.035.  Google Scholar

[10]

L. Jeanjean, Existence of solutions with prescribed norm for semilinear elliptic equations,, Nonlinear Anal. Theory T. M. & A., 28 (1997), 1633.  doi: 10.1016/S0362-546X(96)00021-1.  Google Scholar

[11]

J. H. 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.  Google Scholar

[12]

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

[13]

Y. Li and W. M. Ni, Radial symmetry of positive solutions of nonlinear ellitic equations in $\mathbbR^n$,, Comm. Partial Differential Equations, 18 (1993), 1043.  doi: 10.1080/03605309308820960.  Google Scholar

[14]

G. B. Li, Some properties of weak solutions of nonlinear scalar fields equation,, Ann. Acad. Sci. Fenn. Math., 15 (1990), 27.  doi: 10.5186/aasfm.1990.1521.  Google Scholar

[15]

G. B. Li and H. Y. Ye, Existence of positive solutions for nonlinear Kirchhoff type problems in $\mathbbR^3$ with critical Sobolev exponent,, Math. Methods Appl. Sci., 37 (2014), 2570.  doi: 10.1002/mma.3000.  Google Scholar

[16]

G. B. Li and H. Y. 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.  Google Scholar

[17]

Y. H. Li et al., 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.  Google Scholar

[18]

J. L. Lions, On some questions in boundary value problems of mathmatical physics,, in Contemporary Development in Continuum Mechanics and Partial Differential Equations, 30 (1978), 284.   Google Scholar

[19]

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

[20]

J. Moser, A New proof of de Giorgi's theorem concerning the regularity problem for elliptic differential equations,, Comm. Pure Appl. Math., 13 (1960), 457.  doi: 10.1002/cpa.3160130308.  Google Scholar

[21]

S. I. Pohožaev, A certain class of quasilinear hyperbolic equations,, Mat. Sb. (NS), 96 (1975), 152.   Google Scholar

[22]

P. Tolksdorf, Regularity for some general class of quasilinear elliptic equations,, J. Differential Equations, 51 (1984), 126.  doi: 10.1016/0022-0396(84)90105-0.  Google Scholar

[23]

N. S. Trudinger, On Harnack type inequalities and their application to quasilinear ellitpic equations,, Comm. Pure Appl. Math., 20 (1967), 721.  doi: 10.1002/cpa.3160200406.  Google Scholar

[24]

J. Wang et al., Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth,, J. Differential Equations, 253 (2012), 2314.  doi: 10.1016/j.jde.2012.05.023.  Google Scholar

[25]

M. Willem, Minimax Theorems,, Progress in Nonlinear Differential Equations and their Applications, (1996).  doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[26]

X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in $\mathbbR^N$,, Nonlinear Anal.: Real World Applications, 12 (2011), 1278.  doi: 10.1016/j.nonrwa.2010.09.023.  Google Scholar

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