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

Abstract

Abstract:

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.

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

CLC Number:

35J50

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.

Trendmd

References

[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

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 15

Metrics

Comments

Recommended 0

[1]	Pengyan WANG, Yongzhong WANG. POSITIVE SOLUTIONS FOR A WEIGHTED FRACTIONAL SYSTEM [J]. Acta mathematica scientia,Series B, 2018, 38(3): 935-949.
[2]	Dexing KONG, Qi LIU. GLOBAL EXISTENCE OF CLASSICAL SOLUTIONS TO THE HYPERBOLIC GEOMETRY FLOW WITH TIME-DEPENDENT DISSIPATION [J]. Acta mathematica scientia,Series B, 2018, 38(3): 745-755.
[3]	Lei ZHANG, Bin LIU. THE GLOBAL ATTRACTOR FOR A VISCOUS WEAKLY DISSIPATIVE GENERALIZED TWO-COMPONENT μ-HUNTER-SAXTON SYSTEM [J]. Acta mathematica scientia,Series B, 2018, 38(2): 651-672.
[4]	Jing ZHANG, Yongqian ZHANG. INITIAL-BOUNDARY PROBLEM FOR THE 1-D EULER-BOLTZMANN EQUATIONS IN RADIATION HYDRODYNAMICS [J]. Acta mathematica scientia,Series B, 2018, 38(1): 34-56.
[5]	Yingbo LIU, Ingo WITT. SMALL DATA SOLUTIONS OF 2-D QUASILINEAR WAVE EQUATIONS UNDER NULL CONDITIONS [J]. Acta mathematica scientia,Series B, 2018, 38(1): 125-150.
[6]	Biao WANG. POSITIVE STEADY STATES OF A DIFFUSIVE PREDATOR-PREY SYSTEM WITH PREDATOR CANNIBALISM [J]. Acta mathematica scientia,Series B, 2017, 37(5): 1385-1398.
[7]	Wei DAI, Zhao LIU. CLASSIFICATION OF POSITIVE SOLUTIONS TO A SYSTEM OF HARDY-SOBOLEV TYPE EQUATIONS [J]. Acta mathematica scientia,Series B, 2017, 37(5): 1415-1436.
[8]	Fei HOU. ON THE GLOBAL EXISTENCE OF SMOOTH SOLUTIONS TO THE MULTI-DIMENSIONAL COMPRESSIBLE EULER EQUATIONS WITH TIME-DEPENDING DAMPING IN HALF SPACE [J]. Acta mathematica scientia,Series B, 2017, 37(4): 949-964.
[9]	Anyin XIA, Mingshu FAN, Shan LI. BLOW-UP AND LIFE SPAN ESTIMATES FOR A CLASS OF NONLINEAR DEGENERATE PARABOLIC SYSTEM WITH TIME-DEPENDENT COEFFICIENTS [J]. Acta mathematica scientia,Series B, 2017, 37(4): 974-984.
[10]	Yanyan GUO. NONEXISTENCE AND SYMMETRY OF SOLUTIONS TO SOME FRACTIONAL LAPLACIAN EQUATIONS IN THE UPPER HALF SPACE [J]. Acta mathematica scientia,Series B, 2017, 37(3): 836-851.
[11]	Zhenhai LIU, Shengda ZENG. DIFFERENTIAL VARIATIONAL INEQUALITIES IN INFINITE BANACH SPACES [J]. Acta mathematica scientia,Series B, 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]. Acta mathematica scientia,Series B, 2017, 37(1): 47-57.
[13]	Mohammed D. KASSIM, Khaled M. FURATI, Nasser-eddine TATAR. NON-EXISTENCE FOR FRACTIONALLY DAMPED FRACTIONAL DIFFERENTIAL PROBLEMS [J]. Acta mathematica scientia,Series B, 2017, 37(1): 119-130.
[14]	Li XIA, Jingna LI, Qiang LIU. EXISTENCE OF WEAK SOLUTIONS FOR SOME SINGULAR PARABOLIC EQUATION [J]. Acta mathematica scientia,Series B, 2016, 36(6): 1651-1661.
[15]	Li WEI, Rui CHEN, Ravi P. AGARWAL, Patricia YJ WONG. EXISTENCE AND UNIQUENESS OF NON-TRIVIAL SOLUTION OF PARABOLIC p-LAPLACIAN-LIKE DIFFERENTIAL EQUATION WITH MIXED BOUNDARIES [J]. Acta mathematica scientia,Series B, 2016, 36(6): 1780-1792.