PDF(579 KB)
PDF(579 KB)
PDF(579 KB)
一类具有非局部项和临界项椭圆方程的正解
Positive Solution for a Class of Elliptic Equations Involving Nonlocal Term and Critical Term
本文研究下列具有临界项的Kirchhoff型方程（a+b∫R3[|∇u|2+V（x）u2]dx）·[-△u+V（x）u]=μf（x，u）+K（x）u5，x∈R3，其中a，b，μ>0，位势函数V，K满足一些恰当的条件，非线性项f满足超三次或超线性增长性条件.利用山路定理，得到三个存在性结果.
In the paper, the following Kirchhoff-type equations with critical exponent (a+b∫R3[|∇u|2+V(x)u2]dx)·[-△u+V(x)u]=μf(x,u)+K(x)u5,x∈R3, is studied, in which a, b, μ > 0, the potential functions V, K satisfy some conditions and the nonlinear term f verifies sup-cubic or sup-linear growth. With the aid of the mountain pass theorem, three existence results are obtained.
Kirchhoff型方程 / 山路定理 / 临界指数 {{custom_keyword}} /
Kirchhoff-type equations / Mountain pass theorem / critical exponent {{custom_keyword}} /
[1] Alves C. O., Correa F. J. S. A., Figueiredo G. M., On a class of nonlocal elliptic problems with critical growth, Differ. Equ. Appl., 2010, 2(3):409-417.
[2] Benci V., Cerami G., Existence of positive solutions of the equation -Δu+a(x)u=u(N+2)/(N-2) in RN, J. Funct. Anal., 1990, 88(1):90-117.
[3] Brezis H., Nirenberg, L., Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 1983, 36(4):437-477.
[4] Cerami G., Fortunato D., Struwe M., Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1984, 1(5):341-350.
[5] Chabrowski J., Concentration-compactness principle at infinity and semilinear elliptic equations involving critical and subcritical Sobolev exponents, Calc. Var. Partial Differential Equations, 1995, 3(4):493-512.
[6] Egnell H., Semilinear elliptic equations involving critical Sobolev exponents, Arch. Rational Mech. Anal., 1988, 104(1):27-56.
[7] Escobar J. F., Positive solutions for some semilinear elliptic equations with critical Sobolev exponents, Comm. Pure Appl. Math., 1987, 40(5):623-657.
[8] He X., Zou, W., Ground states for nonlinear Kirchhoff equations with critical growth, Ann. Mat. Pura Appl., 2014, 193(2):473-500.
[9] He Y., Li G., Peng S., Concentrating bound states for Kirchhoff type problems in R3 involving critical Sobolev exponents, Adv. Nonlinear Stud., 2014, 14(2):483-510.
[10] Jeanjean L., Le Coz S., An existence and stability result for standing waves of nonlinear Schrödinger equations, Adv. Differential Equations, 2006, 11(7):813-840.
[11] Kikuchi H., Existence and stability of standing waves for Schrödinger-Poisson-Slater equation, Adv. Nonlinear Stud., 2007, 7(3):403-437.
[12] Lei C. Y., Liao J. F., Tang C. L., Multiple positive solutions for Kirchhoff type of problems with singularity and critical exponents, J. Math. Anal. Appl., 2015, 421(1):521-538.
[13] Li Y., Li F., Shi J., Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differential Equations, 2012, 253(7):2285-2294.
[14] Lins H. F., Silva E. A. B., Quasilinear asymptotically periodic elliptic equations with critical growth, Nonlinear Anal., 2009, 71(7/8):2890-2905.
[15] Lions P. L., The concentration-compactness principle in the calculus of variations, The limit case. I, Rev. Mat. Iberoamericana, 1985, 1(1):145-201.
[16] Miyagaki O. H., On a class of semilinear elliptic problems in RN with critical growth, Nonlinear Anal., 1997, 29(7):773-781.
[17] Naimen D., Positive solutions of Kirchhoff type elliptic equations involving a critical Sobolev exponent, NoDEA Nonlinear Differential Equations Appl., 2014, 21(6):885-914.
[18] Naimen D., On the Brezis-Nirenberg problem with a Kirchhoff type perturbation, Adv. Nonlinear Stud., 2015, 15(1):135-156.
[19] Sun Y., Liu X., Existence of positive solutions for Kirchhoff type problems with critical exponent, J. Partial Differ. Equ., 2012, 25(2):187-198.
[20] Wang J., Tian L., Xu J., Zhang F., Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differential Equations, 2012, 253(7):2314-2351.
[21] Xie Q. L., Wu X. P., Tang C. L., Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent, Commun. Pure Appl. Anal., 2013, 12(6):2773-2786.
[22] Ye Y., Tang C. L., Multiple solutions for Kirchhoff-type equations in RN, J. Math. Phys., 2013, 54(8):081508, 16 pp.
[23] Zhang H., Zhang F., Ground states for the nonlinear Kirchhoff type problems, J. Math. Anal. Appl., 2015, 423(2):1671-1692.
国家自然科学基金（11471267）；贵州省教育厅自然科学研究项目（黔教合KY字[2015]453）；黔南民族师范学院非线性分析科研创新团队（Qnsyk201606）
/
| 〈 |
|
〉 |
