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:
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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.

[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.

[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.

[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.

[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.

[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.

[7]

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

[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.

[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.

[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.

[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.

[12]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

[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.

[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.

[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.

[25]

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

[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.

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.

[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.

[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.

[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.

[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.

[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.

[7]

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

[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.

[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.

[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.

[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.

[12]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

[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.

[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.

[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.

[25]

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

[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.

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