THE EXISTENCE OF NONTRIVIAL SOLUTIONS TO A SEMILINEAR ELLIPTIC SYSTEM ON RN WITHOUT THE AMBROSETTI-RABINOWITZ CONDITION

doi:10.1016/S0252-9602(10)60183-X

数学物理学报(英文版) ›› 2010, Vol. 30 ›› Issue (6): 1917-1936.doi: 10.1016/S0252-9602(10)60183-X

THE EXISTENCE OF NONTRIVIAL SOLUTIONS TO A SEMILINEAR ELLIPTIC SYSTEM ON R^N WITHOUT THE AMBROSETTI-RABINOWITZ CONDITION

李工宝|王春花

School of Mathematics and Statistics, |Huazhong Normal University, Wuhan 430079, |China

收稿日期:2010-07-02 出版日期:2010-11-20 发布日期:2010-11-20
通讯作者: Gongbao Li; E-mail: ligb@mail.ccnu.edu.cn E-mail:ligb@mail.ccnu.edu.cn
基金资助:
Partially supported by NSFC (10571069, 10631030) and Hubei Key Laboratory of Mathematical Sciences.

Partially supported by the fund of CCNU for PHD students(2009019).

THE EXISTENCE OF NONTRIVIAL SOLUTIONS TO A SEMILINEAR ELLIPTIC SYSTEM ON R^N WITHOUT THE AMBROSETTI-RABINOWITZ

LI Gong-Bao, WANG Chun-Hua

School of Mathematics and Statistics, |Huazhong Normal University, Wuhan 430079, |China

Received:2010-07-02 Online:2010-11-20 Published:2010-11-20
Contact: Gongbao Li; E-mail: ligb@mail.ccnu.edu.cn E-mail:ligb@mail.ccnu.edu.cn
Supported by:
Partially supported by NSFC (10571069, 10631030) and Hubei Key Laboratory of Mathematical Sciences.
Partially supported by the fund of CCNU for PHD students(2009019).

摘要/Abstract

摘要：

In this paper, we prove the existence of at least one positive solution pair (u, v)∈H¹(R^N)×H¹(R^N) to the following semilinear elliptic system

{-Δu+u=f(x, v), x∈R^N,
-Δv+v=g(x, u), x∈R^N, (0.1)

by using a linking theorem and the concentration-compactness principle. The main conditions we imposed on the nonnegative functions f, g ∈ C⁰(R^N×R¹) are that, f(x, t ) and g(x, t) are superlinear at t=0 as well as at t=+∞, that f and g are subcritical in t and satisfy a kind of monotonic conditions. We mention that we do not assume that f or g satisfies the Ambrosetti-Rabinowitz condition as usual.

Our main result can be viewed as an extension to a recent result of Miyagaki and Souto [J. Diff. Equ. 245(2008), 3628-3638] concerning the existence of a positive solution to the semilinear elliptic boundary value problem

{-Δu+u=f(x, u), x∈Ω,
u∈H¹₀(Ω)
where Ω（R^N is bounded and a result of Li and Yang [G. Li and J. Yang: Communications in P.D.E. Vol. 29(2004) Nos.5 & 6.pp.925--954, 2004] concerning (0.1) when f and g are asymptotically linear.

关键词: existence, nontrivial solution, semilinear elliptic system, without the (AR) condition

Abstract:

Key words: existence, nontrivial solution, semilinear elliptic system, without the (AR) condition

中图分类号:

35J50

李工宝, 王春花. THE EXISTENCE OF NONTRIVIAL SOLUTIONS TO A SEMILINEAR ELLIPTIC SYSTEM ON R^N WITHOUT THE AMBROSETTI-RABINOWITZ CONDITION[J]. 数学物理学报(英文版), 2010, 30(6): 1917-1936.

LI Gong-Bao, WANG Chun-Hua. THE EXISTENCE OF NONTRIVIAL SOLUTIONS TO A SEMILINEAR ELLIPTIC SYSTEM ON R^N WITHOUT THE AMBROSETTI-RABINOWITZ[J]. Acta mathematica scientia,Series B, 2010, 30(6): 1917-1936.

参考文献

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

[2] Berestycki H, Lions P L. Nonlinear scalar field equations-I and II. Arch Rational Mech Anal, 1983, 82: 313--376

[3] Strass W A. Existence of solitary waves in higher dimensions. Comm Math Phys, 1997, 55: 149--156

[4] Costa D G. On a class of elliptic systems in R^N. Elect J Differ Equ, 1994, 7: 1--14

[5] Clement P, Figuereido D G, Mitidieri E. Positive solutions of semilinear elliptic syestems. Comm Partial Differ Equ, 1992, 17: 923--940

[6] Lions P L. The concentration-compcatness principle in the calculus of variations, The locally compact case. Ann I H P Analyse Nonlin, 1984, 1(4): 223--282

[7] Van De Vorst R C A M. Variational identities and applications to differential systems. Arch Rational Mech Anal, 1991, 116: 375--398

[8] Hulshof J, Van De Vorst R C A M. Differential systems with strongly indefinite varational structure. J Funct Anal, 1993, 114: 32--58

[9] De Finueirdo D G, Felmer P L. On superquadratic elliptic systems. Tras Amer Math Soc, 1994, 343: 99--116

[10] 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

[11] Ding Y H, Li S J. Existence of entire solutions for some elliptic systems. Bull Austral Math Soc, 1994, 19: 501--519

[12] Reed M, Simon B. Methods of modern mathematics pysics, IV. New York, San Francisco, London Academic Press, 1978

[13] Busca J, Sirakov B. Symmetry results for semilinear elliptic systems in he whole space. J. Diff. Equas, 2000，163: 41-56

[14] Jeanjean L. On the existence of bounded Palais-Smale sequences and appilcation to a Landesmann-Laze type problem. Proc Royal Soc Ediub A, 1999, 129: 787--809

[15] Li G B, Zhou H S. The existence of a positive solutions to asymptotically linear scalar field equations. Proc Royal Soc Ediub A, 2000, 130: 81--105

[16] Li G B, Szulkin A. An asymptically periodic Schr\"{o}dinger equation with indefinite linear part. Comm Contemp Math, 2002, 4: 763--776

[17] Li G B, Yang J F. Asymptotically linear elliptic systems. Comm Partial Differ Equ, 2004, 29(5/6): 925--954

[18] Miyagaki O H, Souto M A S. Super-linear problems without Ambrosetti and Rabinowitz growth condition. J Differential Equations, 2008, 245: 3628--3638

[19] Yang J H, Zhu X P. On the existence of nontrivial solution of a quasilinear elliptic boundary value problem for unbounded domains. I. Acta Math Sci, 1987, 7: 341--359

[20] Kryszewski W, Szulkin A. Generalized linking theorem with an application to semilinear Schr\"{o}dinger equation. Adv Differ Equ, 1998, 3: 441--472

[21] Willem M. Minimax Theorems. Boston, Basel, Berlin: Birkh\"auser, 1996

THE EXISTENCE OF NONTRIVIAL SOLUTIONS TO A SEMILINEAR ELLIPTIC SYSTEM ON R^N WITHOUT THE AMBROSETTI-RABINOWITZ CONDITION

THE EXISTENCE OF NONTRIVIAL SOLUTIONS TO A SEMILINEAR ELLIPTIC SYSTEM ON R^N WITHOUT THE AMBROSETTI-RABINOWITZ

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 0

[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]	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.
[4]	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.
[5]	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.
[6]	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.
[7]	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.
[8]	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.
[9]	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.
[10]	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.
[11]	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.
[12]	Dragos-Patru COVEI. A NECESSARY AND A SUFFICIENT CONDITION FOR THE EXISTENCE OF THE POSITIVE RADIAL SOLUTIONS TO HESSIAN EQUATIONS AND SYSTEMS WITH WEIGHTS[J]. 数学物理学报(英文版), 2017, 37(1): 47-57.
[13]	Mohammed D. KASSIM, Khaled M. FURATI, Nasser-eddine TATAR. NON-EXISTENCE FOR FRACTIONALLY DAMPED FRACTIONAL DIFFERENTIAL PROBLEMS[J]. 数学物理学报(英文版), 2017, 37(1): 119-130.
[14]	夏莉, 李敬娜, 刘强. EXISTENCE OF WEAK SOLUTIONS FOR SOME SINGULAR PARABOLIC EQUATION[J]. 数学物理学报(英文版), 2016, 36(6): 1651-1661.
[15]	魏利, 陈蕊, Ravi P. AGARWAL, Patricia YJ WONG. EXISTENCE AND UNIQUENESS OF NON-TRIVIAL SOLUTION OF PARABOLIC p-LAPLACIAN-LIKE DIFFERENTIAL EQUATION WITH MIXED BOUNDARIES[J]. 数学物理学报(英文版), 2016, 36(6): 1780-1792.