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

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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

Abstract
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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.

Keywords: Kirchhoff equation, barycenter map., Nehari-Pohožaev type manifold, superlinear nonlinearities.

Mathematics Subject Classification: 35J60, 35A1.

Citation: Hongyu Ye. Positive high energy solution for Kirchhoff equation in $\mathbb{R}^{3}$ with superlinear nonlinearities via Nehari-Pohožaev manifold. Discrete and Continuous Dynamical Systems, 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), 43-56.
[2]	P. D'Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math., 108 (1992), 247-262. doi: 10.1007/BF02100605.
[3]	A. Arosio and S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc., 348 (1996), 305-330. 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), 981-1012. 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), 259-281. 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), 827-850. 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), 17-26.
[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), 701-730.
[9]	X. M. He and W. M. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in $\mathbb{R}^3$, J. Differential Equations, 252 (2012), 1813-1834. 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), 1633-1659. doi: 10.1016/S0362-546X(96)00021-1.
[11]	J. H. Jin and X. Wu, Infinitely many radial solutions for Kirchhoff-type problems in $\mathbb{R}^N2$, J. Math. Anal. Appl., 369 (2010), 564-574. 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 $\mathbb{R}^{N}$, Comm. Partial Differential Equations, 18 (1993), 1043-1054. 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), 27-36. doi: 10.5186/aasfm.1990.1521.
[15]	G. B. Li and H. Y. Ye, Existence of positive solutions for nonlinear Kirchhoff type problems in $\mathbb{R}^3$ with critical Sobolev exponent, Math. Methods Appl. Sci., 37 (2014), 2570-2584. 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 $\mathbb{R}^3$, J. Differential Equations, 257 (2014), 566-600. 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), 2285-2294. 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, in North-Holland Math. Stud., North-Holland, Amsterdam, New York, 30 (1978), 284-346.
[19]	W. Liu and X. M. 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.
[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), 457-468. doi: 10.1002/cpa.3160130308.
[21]	S. I. Pohožaev, A certain class of quasilinear hyperbolic equations, Mat. Sb. (NS), 96 (1975), 152-166, 168 (in Russian).
[22]	P. Tolksdorf, Regularity for some general class of quasilinear elliptic equations, J. Differential Equations, 51 (1984), 126-150. 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), 721-747. 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), 2314-2351. doi: 10.1016/j.jde.2012.05.023.
[25]	M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 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 $\mathbb{R}^N2$, Nonlinear Anal.: Real World Applications, 12 (2011), 1278-1287. doi: 10.1016/j.nonrwa.2010.09.023.

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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), 43-56.
[2]	P. D'Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math., 108 (1992), 247-262. doi: 10.1007/BF02100605.
[3]	A. Arosio and S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc., 348 (1996), 305-330. 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), 981-1012. 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), 259-281. 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), 827-850. 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), 17-26.
[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), 701-730.
[9]	X. M. He and W. M. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in $\mathbb{R}^3$, J. Differential Equations, 252 (2012), 1813-1834. 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), 1633-1659. doi: 10.1016/S0362-546X(96)00021-1.
[11]	J. H. Jin and X. Wu, Infinitely many radial solutions for Kirchhoff-type problems in $\mathbb{R}^N2$, J. Math. Anal. Appl., 369 (2010), 564-574. 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 $\mathbb{R}^{N}$, Comm. Partial Differential Equations, 18 (1993), 1043-1054. 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), 27-36. doi: 10.5186/aasfm.1990.1521.
[15]	G. B. Li and H. Y. Ye, Existence of positive solutions for nonlinear Kirchhoff type problems in $\mathbb{R}^3$ with critical Sobolev exponent, Math. Methods Appl. Sci., 37 (2014), 2570-2584. 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 $\mathbb{R}^3$, J. Differential Equations, 257 (2014), 566-600. 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), 2285-2294. 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, in North-Holland Math. Stud., North-Holland, Amsterdam, New York, 30 (1978), 284-346.
[19]	W. Liu and X. M. 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.
[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), 457-468. doi: 10.1002/cpa.3160130308.
[21]	S. I. Pohožaev, A certain class of quasilinear hyperbolic equations, Mat. Sb. (NS), 96 (1975), 152-166, 168 (in Russian).
[22]	P. Tolksdorf, Regularity for some general class of quasilinear elliptic equations, J. Differential Equations, 51 (1984), 126-150. 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), 721-747. 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), 2314-2351. doi: 10.1016/j.jde.2012.05.023.
[25]	M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 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 $\mathbb{R}^N2$, Nonlinear Anal.: Real World Applications, 12 (2011), 1278-1287. doi: 10.1016/j.nonrwa.2010.09.023.

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