THE EXISTENCE OF SOLUTIONS OF QUASILINEAR ELLIPTIC EQUATIONS WITH CHANGE OF SIGN

数学物理学报(英文版) ›› 2001, Vol. 21 ›› Issue (4): 469-482.

THE EXISTENCE OF SOLUTIONS OF QUASILINEAR ELLIPTIC EQUATIONS WITH CHANGE OF SIGN

李工宝, 余纯

Young Scientist Laboratory of Mathematical Sciences, Wuhan Institute of Physics and Mathematics,
Chinese Academy of Sciences, P.O.Box 71010, Wuhan 430071, China Department of Mathematics, Wuhan University, Wuhan 430072, China

出版日期:2001-10-06 发布日期:2001-10-06
基金资助:
The first author is partially supported by NSFC and Academy of Finland.

THE EXISTENCE OF SOLUTIONS OF QUASILINEAR ELLIPTIC EQUATIONS WITH CHANGE OF SIGN

LI Gong-Bao, YU Chun

Young Scientist Laboratory of Mathematical Sciences, Wuhan Institute of Physics and Mathematics,
Chinese Academy of Sciences, P.O.Box 71010, Wuhan 430071, China Department of Mathematics, Wuhan University, Wuhan 430072, China

Online:2001-10-06 Published:2001-10-06
Supported by:
The first author is partially supported by NSFC and Academy of Finland.

摘要/Abstract

摘要：

This paper considers the following quasilinear elliptic problem( −div(|∇u|p−2∇u) = a(x)g(u) in u = 0 on @ where is a bounded regular domain in RN(N≥3), N > p > 1. When g(u) satisfies suitable conditions and g(u)u − R u 0g(s)ds is unbounded, a(x) is a H¨older continuous function which changes sign on and R − |a(x)|dx is suitably small. The authors prove the existence of a nonnegative nontrivial solution for N > p > 1, in particular, the existence of a positive solution to the problem for N > p≥2. Our main theorem generalizes a recent result of Samia Khanfir and Leila Lassoued (see [1]) concerning the case where p = 2.They prove also that if g(u) = |u|q−2u with p < q < p and + = {x∈|a(x) > 0} is a nonempty open set, then the above problem possesses infinitely many solutions.

关键词: Qasilinear elliptic equation, (PS) condition, mountain-pass Lemma, infinite solution

Abstract:

Key words: Qasilinear elliptic equation, (PS) condition, mountain-pass Lemma, infinite solution

中图分类号:

35J60

李工宝, 余纯. THE EXISTENCE OF SOLUTIONS OF QUASILINEAR ELLIPTIC EQUATIONS WITH CHANGE OF SIGN[J]. 数学物理学报(英文版), 2001, 21(4): 469-482.

LI Gong-Bao, YU Chun. THE EXISTENCE OF SOLUTIONS OF QUASILINEAR ELLIPTIC EQUATIONS WITH CHANGE OF SIGN[J]. Acta mathematica scientia,Series B, 2001, 21(4): 469-482.

参考文献

[1]Samia Khanfir, Leila Lassoued. On the existence of positive solutions of a semilinear elliptic equation with change of sign, Nonlinear Analysis, Theory, Methods, Aplications, 1994, 22(11): 1039-1314

[2]Ambrosetti a A, Rabinowitz P H. Dual variational methods incritical point theory and application. J Funct Anal, 1973, 14: 349-381

[3]Shen Yaotian. The nontrivial solutions of quasilinear elliptic equations in W1,p0 . Scientia Sinica (Ser A),1984,27: 720-730

[4]Vazquez J L. A strong maximum principle for some quasilinear elliptic equations. Appl Math Optim,1984,12: 191-202

[5]Guedda M, Veron L. Qusilinear elliptic equations involving critical Sololev exponents. Nonlinear Analysis,Theory, Methods, Applications, 1989, 13(8): 879-902

[6]Li Gongbao. Some properties of weak solutions of nonlinear scalar field equations. Annales Academiae Scientiarum Fennicae Series A I Mathematica, 1989,14: 27-36

[7]Gilbarg D, Trudinger N S. Elliptic partial differential equations of second order. Heidelberg, New York:Springer-Verlag, 1983

[8]Peter Tolksdorf. Regularity for a more general class of quasilinear elliptic equations. J Diff Equs,1984,51:121-160

[9]Robinowitz P H. Minimax methods in critical point theory with applications to differential equations. Regional Conference Series in Mathematics, ISSN 0160-7642, no.65,AMS,1986

[10]Li Gongbao. The existence of infinitely many solutions of quasilinear partial differential equations in unbounded domains. Acta Math Sci, 1989,9(2):175-188

[11]Li Gongbao, Zhu Xiping. Nontrivial solution of nonlinear scalar field equations with strong nonlinearity.Acta Math Sci, 1988, 8(4): 431-448

[12]Li Gongbao, Olli Martio. Stability in obstacle problems. Math Scand, 1994, 75: 87-100

[13]Chen Yiaze, Wu Lanchen. Second order elliptic equations and elliptic systems. Science Press, 1991(Chinese Edition)

THE EXISTENCE OF SOLUTIONS OF QUASILINEAR ELLIPTIC EQUATIONS WITH CHANGE OF SIGN

THE EXISTENCE OF SOLUTIONS OF QUASILINEAR ELLIPTIC EQUATIONS WITH CHANGE OF SIGN

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