EXISTENCE OF POSITIVE SOLUTIONS TO SEMILINEAR ELLIPTIC SYSTEMS IN RN WITH ZERO MASS

doi:10.1016/S0252-9602(13)60050-8

数学物理学报(英文版) ›› 2013, Vol. 33 ›› Issue (4): 913-928.doi: 10.1016/S0252-9602(13)60050-8

EXISTENCE OF POSITIVE SOLUTIONS TO SEMILINEAR ELLIPTIC SYSTEMS IN RN WITH ZERO MASS

李工宝*|叶红雨

School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China

收稿日期:2012-07-05 修回日期:2012-09-27 出版日期:2013-07-20 发布日期:2013-07-20
通讯作者: 李工宝，ligb@mail.ccnu.edu.cn E-mail:ligb@mail.ccnu.edu.cn
基金资助:
Partially supported by NSFC (11071095) and Hubei Key Laboratory of Mathematical Sciences.

EXISTENCE OF POSITIVE SOLUTIONS TO SEMILINEAR ELLIPTIC SYSTEMS IN RN WITH ZERO MASS

LI Gong-Bao*, YE Hong-Yu

School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China

Received:2012-07-05 Revised:2012-09-27 Online:2013-07-20 Published:2013-07-20
Contact: LI Gong-Bao，ligb@mail.ccnu.edu.cn E-mail:ligb@mail.ccnu.edu.cn
Supported by:
Partially supported by NSFC (11071095) and Hubei Key Laboratory of Mathematical Sciences.

摘要/Abstract

摘要：

In this paper, we prove the existence of at least one positive solution pair (u, v) ∈ D^1,2(R^N) × D^1,2(R^N) to the following semilinear elliptic system
{−△u = K(x)f(v), x ∈ R^N,
−△v = K(x)g(u), x ∈ R^N(0.1)
by using a linking theorem, where K(x) is a positive function in Ls(R^N) for some s > 1 and the nonnegative functions f, g ∈ C(R, R) are of quasicritical growth, superlinear at infinity. We do not assume that f or g satisfies the Ambrosetti-Rabinowitz condition as usual.
Our main result can be viewed as a partial extension of a recent result of Alves, Souto and Montenegro in [1] concerning the existence of a positive solution to the following semilinear elliptic problem

−△u = K(x)f(u), x ∈ R^N,

and a recent result of Li and Wang in [22] concerning the existence of nontrivial solutions to a semilinear elliptic system of Hamiltonian type in R^N.

关键词: existence, positive solutions, elliptic system, linking geometric structure, zero mass case

Abstract:

−△u = K(x)f(u), x ∈ R^N,

and a recent result of Li and Wang in [22] concerning the existence of nontrivial solutions to a semilinear elliptic system of Hamiltonian type in R^N.

Key words: existence, positive solutions, elliptic system, linking geometric structure, zero mass case

中图分类号:

35J60

李工宝, 叶红雨. EXISTENCE OF POSITIVE SOLUTIONS TO SEMILINEAR ELLIPTIC SYSTEMS IN RN WITH ZERO MASS[J]. 数学物理学报(英文版), 2013, 33(4): 913-928.

LI Gong-Bao, YE Hong-Yu. EXISTENCE OF POSITIVE SOLUTIONS TO SEMILINEAR ELLIPTIC SYSTEMS IN RN WITH ZERO MASS[J]. Acta mathematica scientia,Series B, 2013, 33(4): 913-928.

参考文献

[1] Alves C O, Souto M A S, Montenegro M. Existence of solution for two classes of elliptic problems in RN with zero mass. J Differ Equ, 2012, 252: 5735–5750

[2] Ambrosetti A, Felli V, Malchiodi A. Ground states of nonlinear Schr¨odinger equations with potentials vanishing at infinity. J Eur Math Soc, 2005, 7: 117–144

[3] Ambrosetti A, Rabinowitz P. Dual variational methods in critical points theory and applications. J Funct Anal, 1973, 14: 349–381

[4] Ambrosetti A, Wang Z Q. Nonlinear Schr¨odinger equations with vanishing and decaying potentials. Differ Integral Equ, 2005, 18: 1321–1332

[5] Aubin T. Probl´ems isop´erim´etriques et espaces de Sobolev. J Differ Geom, 1976, 11: 573–598

[6] Azzollini A, Pomponio A. On a “zero mass” nonlinear Schr¨odinger equation. Adv Nonlinear Stud, 2007, 7: 599–628

[7] Azzollini A, Pomponio A. Compactness results and applications to some “zero mass” elliptic problems. Nonlinear Anal, 2008, 69: 3559–3576

[8] Benci V, Fortunato D. Towards a unified theorey for classical electrodynamics. Arch Rational Mech Anal, 2004, 173: 379–414

[9] Badiale M, Pisani L, Rolando S. Sum of weighted Lebesgue spaces and nonlinear elliptic equations. NoDEA Nonlinear Differential Equations Appl, 2011, 18(4): 369–405

[10] Berestycki H, Lions P L. Nonlinear scalar field equations-I and I. Existence of a ground state. Arch Rational Mech Anal, 1983, 82: 313–345

[11] Berestycki H, Lions P L. Nonlinear scalar field equations-I and II. exitence of infinitely many solutions. Arch Rational Mech Anal, 1983, 82: 347–376

[12] Berestycki H, Lions P L. Existence d’´etats multiples dans des ´equations de champs scalaires non lin´eaires dans le cas masse nulle. C R Acad Sci Paris S´er I Math, 1983, 297: 267–270

[13] Benci V, Grisanti C R, Micheletti A M. Existence and nonexistence of the ground state solution fot the nonlinear Schr¨odinger equations with V (1) = 0. Topol Methods Nonlinear Anal, 2005, 26: 203–219

[14] Benci V, Grisanti C R, Micheletti A M. Existence of solutions for the nonlinear Schr¨odinger equation with V (1) = 0. Progr Nonlinear Equations Appl, 2005, 66: 53–65

[15] Busca J, Sirakov B. Symmetric results for semilinear elliptic systems in the whole space. J Differ Equ, 2000, 163: 41–56

[16] Caffarelli L A, Gidas B, Spruck J. Asymptotic symmetric and local behavior of semilinear elliptic equations with critical Sobolev growth. Comm Pure Appl Math, 1989, 42(3): 271–297

[17] Ding Y H, Luan S X. Multiple solutions for a class of nonlinear Schr¨odinger equations. J Differ Equ, 2004, 207: 423–457

[18] De Finueirdo D G, Yang J F. Decay, symmetry and existence of solutions of semilinear elliptic syestems. Nolinear Anal T M A, 1998, 33: 211–234

[19] Ghiment M, Micheletti A M. Existence of minimal nodal solutions for the nonlinear Schr¨odinger equation with V (1) = 0. Adv Differ Equ, 2006, 11(12): 1375–1396

[20] Kryszewski W, Szulkin A. Generalized linking theorem with an application to semilinear Schr¨odinger equation. Adv Differ Equ, 1998, 3: 441–472

[21] Li G B, Szulkin A. An asymptically periodic Schr¨odinger equation with indefinite linear part. Comm Contemp Math, 2002, 4: 763–776

[22] Li G B, Wang C H. The existence of nontivial solutiond to a semilinear elliptic system on RN without the Ambrosetti-Rabinowitz condition. Acta Math Sci, 2010, 30B(6): 1917–1936

[23] Li G B, Yang J F. Asymptotically linear elliptic systems. Comm Partial Differ Equ, 2004, 29: 925–954

[24] Lions P L. The concentration-compcatness principle in the calculus of variations, The locally compact case II. Ann Inst H Poincar´e Anal Non Lin´eaire, 1984, 1: 223–282

[25] Liu Z L, Wang Z Q. On the Ambrosetti-Rabinowitz superlinear condition. Adv Nonlinear Stud, 2004, 4: 561–572

[26] Miyagaki O H, Souto M A S. Super-linear problems without Ambrosetti and Rabinowitz growth condition. J Differ Equ, 2008, 245: 3628–3638

[27] Mao A, Zhang Z T. Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition. Nonlinear Anal, 2009, 70: 1275–1287

[28] Strauss W A. Existence of solitary waves in higher dimensions. Comm Math Phys, 1997, 55: 149–162

[29] Talenti G. Best constant in Sobolev inequality. Ann Mat Pura Appl, 1976, 110: 353–372

[30] Willem M. Minimax Theorems. Boston, Basel, Berlin: Birkh¨auser, 1996

EXISTENCE OF POSITIVE SOLUTIONS TO SEMILINEAR ELLIPTIC SYSTEMS IN RN WITH ZERO MASS

EXISTENCE OF POSITIVE SOLUTIONS TO SEMILINEAR ELLIPTIC SYSTEMS IN RN WITH ZERO MASS

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

Metrics

本文评价

推荐阅读 10

[1]	王朋燕, 王永忠. POSITIVE SOLUTIONS FOR A WEIGHTED FRACTIONAL SYSTEM[J]. 数学物理学报(英文版), 2018, 38(3): 935-949.
[2]	孔德兴, 刘琦. GLOBAL EXISTENCE OF CLASSICAL SOLUTIONS TO THE HYPERBOLIC GEOMETRY FLOW WITH TIME-DEPENDENT DISSIPATION[J]. 数学物理学报(英文版), 2018, 38(3): 745-755.
[3]	廖家锋, 蒲洋, 唐春雷. MULTIPLICITY OF POSITIVE SOLUTIONS FOR A CLASS OF CONCAVE-CONVEX ELLIPTIC EQUATIONS WITH CRITICAL GROWTH[J]. 数学物理学报(英文版), 2018, 38(2): 497-518.
[4]	贾慧芳, 李工宝. MULTIPLICITY AND CONCENTRATION BEHAVIOUR OF POSITIVE SOLUTIONS FOR SCHRÖDINGER-KIRCHHOFF TYPE EQUATIONS INVOLVING THE p-LAPLACIAN IN R^N[J]. 数学物理学报(英文版), 2018, 38(2): 391-418.
[5]	张磊, 刘斌. THE GLOBAL ATTRACTOR FOR A VISCOUS WEAKLY DISSIPATIVE GENERALIZED TWO-COMPONENT μ-HUNTER-SAXTON SYSTEM[J]. 数学物理学报(英文版), 2018, 38(2): 651-672.
[6]	张晶, 张永前. INITIAL-BOUNDARY PROBLEM FOR THE 1-D EULER-BOLTZMANN EQUATIONS IN RADIATION HYDRODYNAMICS[J]. 数学物理学报(英文版), 2018, 38(1): 34-56.
[7]	刘颖博, Ingo WITT. SMALL DATA SOLUTIONS OF 2-D QUASILINEAR WAVE EQUATIONS UNDER NULL CONDITIONS[J]. 数学物理学报(英文版), 2018, 38(1): 125-150.
[8]	邓志颖, 黄毅生. MULTIPLE SYMMETRIC RESULTS FOR A CLASS OF BIHARMONIC ELLIPTIC SYSTEMS WITH CRITICAL HOMOGENEOUS NONLINEARITY IN R^N[J]. 数学物理学报(英文版), 2017, 37(6): 1665-1684.
[9]	王彪. POSITIVE STEADY STATES OF A DIFFUSIVE PREDATOR-PREY SYSTEM WITH PREDATOR CANNIBALISM[J]. 数学物理学报(英文版), 2017, 37(5): 1385-1398.
[10]	马如云, 高红亮. MULTIPLE POSITIVE SOLUTIONS FOR A CLASS OF SEMIPOSITONE NEUMANN PROBLEMS WITH SINGULAR φ-LAPLACIAN[J]. 数学物理学报(英文版), 2017, 37(5): 1472-1482.
[11]	戴蔚, 刘招. CLASSIFICATION OF POSITIVE SOLUTIONS TO A SYSTEM OF HARDY-SOBOLEV TYPE EQUATIONS[J]. 数学物理学报(英文版), 2017, 37(5): 1415-1436.
[12]	侯飞. ON THE GLOBAL EXISTENCE OF SMOOTH SOLUTIONS TO THE MULTI-DIMENSIONAL COMPRESSIBLE EULER EQUATIONS WITH TIME-DEPENDING DAMPING IN HALF SPACE[J]. 数学物理学报(英文版), 2017, 37(4): 949-964.
[13]	夏安银, 樊明书, 李珊. BLOW-UP AND LIFE SPAN ESTIMATES FOR A CLASS OF NONLINEAR DEGENERATE PARABOLIC SYSTEM WITH TIME-DEPENDENT COEFFICIENTS[J]. 数学物理学报(英文版), 2017, 37(4): 974-984.
[14]	郭艳艳. NONEXISTENCE AND SYMMETRY OF SOLUTIONS TO SOME FRACTIONAL LAPLACIAN EQUATIONS IN THE UPPER HALF SPACE[J]. 数学物理学报(英文版), 2017, 37(3): 836-851.
[15]	刘振海, 曾生达. DIFFERENTIAL VARIATIONAL INEQUALITIES IN INFINITE BANACH SPACES[J]. 数学物理学报(英文版), 2017, 37(1): 26-32.