一类具有变号位势的超二次Kirchhoff方程解的多重性
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摘要
本文研究一类全空间上的Kirchhoff型方程.当非线性项是凹凸混合项且f在无穷远处满足超二次增长时,利用变分方法获得方程解的多重性结果,改进和推广了相关文献中的结论.
In this paper, we consider the Kirchhoff type equation on the whole space. When the nonlinearity involves a combination of convex and concave terms and f satisfies super-quadratic growth at infinity, multiplicity of nontrivial solutions to this problem are obtained via variational methods. Our results improves and generalizes that obtained in the literature.
In this paper, we consider the Kirchhoff type equation on the whole space. When the nonlinearity involves a combination of convex and concave terms and f satisfies super-quadratic growth at infinity, multiplicity of nontrivial solutions to this problem are obtained via variational methods. Our results improves and generalizes that obtained in the literature.
引文
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[3] LUO X, WANG Q F. Existence and asymptotic behavior of high energy normalized solutions for the Kirchhoff type equations in R~3[J]. Nonlinear Anal. RWA., 2017, 33:19-32.
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[5] LI G B, YE H Y. Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in R~3[J]. J. Differential Equations, 2014, 257(2):566-600.
[6] CHEN S J, LI L. Multiple solutions for the nonhomogeneous Kirchhoff equation on RN[J]. Nonlinear Analysis. RWA., 2013, 14(3):1477-1486.
[7] DENG Y B, PENG S J, SHUAI W. Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems in R~3[J]. Journal of Functional Analysis, 2015, 269(11):3500-3527.
[8] GUO Z. Ground states for Kirchhoff equations without compact condition[J]. J. Differential Equations, 2015, 259(7):2884-2902.
[9] YE Y W, TANG C L. Multiple solutions for Kirchhoff-type equations in R~N[J]. Journal of Mathematical Physics, 2013, 54(8):081508.
[10] LIU J, LIAO J F, TANG C L. Existence of ground state solution for a class of Kirchhoff type equation in R~N[J]. Proc. Roy. Soc. Edinburgh Sect. A, 2016, 146.
[11] BARTSCH T, WANG Z Q, WILLEM M. The Dirichlet problem for superlinear elliptic equations[M]//Eds. Chipot M, Quittner P, Handbook of Differential Equations-Stationary Partial Differential Equations, Vol. 2, Amsterdam:Elsevier, 2005:1-5.
[12] WILLEM M. Minimax Theorems[M]. Boston:Springer, 1996.
[13] De FIGUEIREDO D G. Positive Solutions of Semilinear Elliptic Problems[M]. Lecture in Math,vol.957, Berlin:Springer-Verlag, 1982.
[14] COSTA D G. An Invitation to Variational Methods in Differential Equations[M]. Berlin:Birkhauser,2006.
[15] RABINOWITZ P H. Minimax Methods in Critical Point Theory with Applications to Differential Equations[M]. CBMS Reg. Conf. Ser. Math., Rhode Island:American Mathematical Soc., 1986.
[2] CHEN S T, TANG X H. Infinitely many solutions for super-quadratic Kirchhoff-type equations with sign-changing potential[J]. Applied Mathematics Letters, 2017, 67:40-45.
[3] LUO X, WANG Q F. Existence and asymptotic behavior of high energy normalized solutions for the Kirchhoff type equations in R~3[J]. Nonlinear Anal. RWA., 2017, 33:19-32.
[4] WU X. Existence of nontrivial solutions and high energy solutions for Schrodinger-Kirchhoff-type equations in R~N[J]. Nonlinear Anal. RWA., 2011, 12:1278-1287.
[5] LI G B, YE H Y. Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in R~3[J]. J. Differential Equations, 2014, 257(2):566-600.
[6] CHEN S J, LI L. Multiple solutions for the nonhomogeneous Kirchhoff equation on RN[J]. Nonlinear Analysis. RWA., 2013, 14(3):1477-1486.
[7] DENG Y B, PENG S J, SHUAI W. Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems in R~3[J]. Journal of Functional Analysis, 2015, 269(11):3500-3527.
[8] GUO Z. Ground states for Kirchhoff equations without compact condition[J]. J. Differential Equations, 2015, 259(7):2884-2902.
[9] YE Y W, TANG C L. Multiple solutions for Kirchhoff-type equations in R~N[J]. Journal of Mathematical Physics, 2013, 54(8):081508.
[10] LIU J, LIAO J F, TANG C L. Existence of ground state solution for a class of Kirchhoff type equation in R~N[J]. Proc. Roy. Soc. Edinburgh Sect. A, 2016, 146.
[11] BARTSCH T, WANG Z Q, WILLEM M. The Dirichlet problem for superlinear elliptic equations[M]//Eds. Chipot M, Quittner P, Handbook of Differential Equations-Stationary Partial Differential Equations, Vol. 2, Amsterdam:Elsevier, 2005:1-5.
[12] WILLEM M. Minimax Theorems[M]. Boston:Springer, 1996.
[13] De FIGUEIREDO D G. Positive Solutions of Semilinear Elliptic Problems[M]. Lecture in Math,vol.957, Berlin:Springer-Verlag, 1982.
[14] COSTA D G. An Invitation to Variational Methods in Differential Equations[M]. Berlin:Birkhauser,2006.
[15] RABINOWITZ P H. Minimax Methods in Critical Point Theory with Applications to Differential Equations[M]. CBMS Reg. Conf. Ser. Math., Rhode Island:American Mathematical Soc., 1986.