POSITIVE SOLUTIONS TO THE p-LAPLACIAN WITH SINGULAR WEIGHTS

doi:10.1016/S0252-9602(12)60075-7

数学物理学报(英文版) ›› 2012, Vol. 32 ›› Issue (3): 1002-1020.doi: 10.1016/S0252-9602(12)60075-7

POSITIVE SOLUTIONS TO THE p-LAPLACIAN WITH SINGULAR WEIGHTS

王影^1*|罗党¹|王明新²

1.School of Mathematics and Information Science, North China University of Water Resources and Electric Power, Zhengzhou 450011, China； 2.Natural Science Research Center, Harbin Institute of Technology, Harbin 150080, China

收稿日期:2010-09-26 出版日期:2012-05-20 发布日期:2012-05-20
通讯作者: 王影,feelyeey@163.com E-mail:feelyeey@163.com
基金资助:
This work was supported by the National Natural Science Foundation of China 10771032.

POSITIVE SOLUTIONS TO THE p-LAPLACIAN WITH SINGULAR WEIGHTS

WANG Ying^1*, LUO Dang¹, WANG Ming-Xin²

1.School of Mathematics and Information Science, North China University of Water Resources and Electric Power, Zhengzhou 450011, China； 2.Natural Science Research Center, Harbin Institute of Technology, Harbin 150080, China

Received:2010-09-26 Online:2012-05-20 Published:2012-05-20
Contact: WANG Ying,feelyeey@163.com E-mail:feelyeey@163.com
Supported by:
This work was supported by the National Natural Science Foundation of China 10771032.

摘要/Abstract

摘要：

By the Mountain Pass Theorem, we study existence and multiplicity of posi-tive solutions of p-laplacian equation of the form −Δ_pu = λf(x, u), the nonlinearity f(x, u) grows as u^σ at infinity with a singular coefficient, where σ∈ (p − 1, p*− 1). To manage the asymptotic behavior of its positive solutions with respect to λ, we establish a new Liouville-type theorem for the p-Laplacian operator.

关键词: p-Laplacian, variational methods, Sobolev-Hardy exponents

Abstract:

Key words: p-Laplacian, variational methods, Sobolev-Hardy exponents

中图分类号:

35J15

王影, 罗党, 王明新. POSITIVE SOLUTIONS TO THE p-LAPLACIAN WITH SINGULAR WEIGHTS[J]. 数学物理学报(英文版), 2012, 32(3): 1002-1020.

WANG Ying, LUO Dang, WANG Ming-Xin. POSITIVE SOLUTIONS TO THE p-LAPLACIAN WITH SINGULAR WEIGHTS[J]. Acta mathematica scientia,Series B, 2012, 32(3): 1002-1020.

参考文献

[1] Ambrosetti A, Brezis H, Cerami G. Combined effects of concave and convex nonlinearities in some elliptic problems. J Funct Anal, 1994, 122(2): 519–543

[2] De Figueiredo D G, Gossez J P, Ubilla P. Local superlinearity and sublinearity for indefinite semilinear elliptic problems. J Funct Anal, 2003, 199(2): 452–467

[3] De Figueiredo D G, Lions P L, Nussbaum R D. A priori estimates and existence of positive solutions of semilinear elliptic equations. J Math Pures Appl, 1982, 61(1): 41–63

[4] Ambrosetti A, Garcia Azorero J, Peral I. Multiplicity results for some nonlinear elliptic equations. J Funct Anal, 1996, 137(1): 219–242

[5] Prashanth S, Sreenadh K. Multiplicity results in a ball for p-Laplace equation with positive nonlinearity. Adv Differential Equations, 2002, 7(7): 877–896

[6] Lions P L. On the existence of positive solutions of semilinear elliptic equations. SIAM Rev, 1982, 24(4): 441–467

[7] Liu Z. Positive solutions of superlinear elliptic equations. J Funct Anal, 1999, 167(2): 370–398

[8] Iturriaga L, Massa E, S´anchez J, Ubilla P. Positive solutions of the p-Laplacian involving a superlinear nonlinearity with zeros. J Differential Equations, 2010, 248: 309–327

[9] Cao D, Han P. Solutions for semilinear elliptic equations with critical exponents and Hardy potential. J Differential Equations, 2004, 205: 521–537

[10] Guoussoub N, Robert F. Concentration estimates for Emden-Fowler equations with boundary singularities and critical growth. Int Math Res Pap IMRP, 2006, Article ID 21867, 1–85

[11] Kang D, Peng S. Solutions for semilinear elliptic problems with critical Sobolev-Hardy exponents and Hardy potential. Appl Math Lett, 2005, 18: 1094–1100

[12] Cao D, Kang D. Solutions of quasilinear elliptic problems involving a Sobolev exponent and multiple Hardy-type terms. J Math Anal Appl, 2007, 333: 889–903

[13] Ghoussoub N, Yuan C. Multiple solutions for quasilinear PDEs involving the critical Sobolev and Hardy exponents. Trans Amer Math Soc, 2000, 352(12): 5703–5743

[14] Han P. Quasilinear elliptic problems with critical exponents and Hardy terms. Nonlinear Anal, 2005, 61: 735–758

[15] Catrina F, Wang Z Q. On the Caffarelli-Kohn-Nirenberg inequalities: Sharp constants, existence (and nonexistence) and symmetry of extremal functions. Comm Pure Appl Math, 2001, 54: 229–258

[16] Felli V, Terracini S. Elliptic equations with multi-singular inverse-square potentials and critical nonlinearity. Comm Partial Differential Equations, 2006, 31: 469–495

[17] Li J. Equation with critical Sobolev-Harfy exponents. Int J Math Sci, 2005, 20: 3213–3223

POSITIVE SOLUTIONS TO THE p-LAPLACIAN WITH SINGULAR WEIGHTS

POSITIVE SOLUTIONS TO THE p-LAPLACIAN WITH SINGULAR WEIGHTS

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