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Continuous averaging proof of the Nekhoroshev theorem
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 |
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.
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] | |
[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. |
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), 43-56. |
[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] | |
[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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