Articles

MULTIPLICITY OF POSITIVE SOLUTIONS FOR SINGULAR ELLIPTIC SYSTEMS WITH CRITICAL SOBOLEV-HARDY AND CONCAVE EXPONENTS

  • Tsing-San Hsu ,
  • Huei-Lin Li
Expand
  • Center for General Education, Chang Gung University, Tao-Yuan, Taiwan, China

Received date: 2009-06-29

  Online published: 2011-05-20

Abstract

In this paper, we consider a singular elliptic system with both concave nonlinearities and critical Sobolev-Hardy growth terms in bounded domains. By means of variational methods, the multiplicity of positive solutions to this problem is obtained.

Cite this article

Tsing-San Hsu , Huei-Lin Li . MULTIPLICITY OF POSITIVE SOLUTIONS FOR SINGULAR ELLIPTIC SYSTEMS WITH CRITICAL SOBOLEV-HARDY AND CONCAVE EXPONENTS[J]. Acta mathematica scientia, Series B, 2011 , 31(3) : 791 -804 . DOI: 10.1016/S0252-9602(11)60276-2

References

[1]  Abdellaoui B, Felli V, Peral I. Existence and multiplicity for perturbations of an equation involving Hardy inequality and critical Sobolev exponent in the whole RN. Adv Differ  Equ, 2004, 9: 481--508

[2]  Alves C O, de Morais Filho D C, Souto M A S. On systems of elliptic equations involving subcritical or critical Sobolev exponents.  Nonlinear Anal,  2000, 42:  771--787

[3]  Ambrosetti A, Garcia J, Peral I. Multiplicity results for some nonlinear elliptic equations. J Funct Anal, 1996, 137: 219--242

[4] Barstch T, Willem M. On a elliptic equation with concave and convex nonlinearities. Proc Amer Math Soc, 1995, 123:  3555--3561

[5]  Brèzis H, Lieb E. A relation between pointwise convergence of functions and convergence of functionals. Proc Amer Math Soc, 1983, 88:  486--490

[6]  Brèzis H, Nirenberg L. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun Pure Appl Math, 1983, 36:  437--477

[7] Brown K J, Zhang Y.  The Nehari manifold for a semilinear elliptic equation with a sign-changing weigh function. J Differ Equ, 2003, 193: 481--499

[8]  Brown  K J,  Wu T F. A semilinear elliptic system involving nonlinear boundary condition and sign-changing weigh function. J Math Anal Appl, 2008, 337:  1326--1336

[9]  Cao D,  Han P. Solutions for semilinear elliptic equations with critical exponents and Hardy potentials. J Differ Equ, 2004, 205: 521--537

[10]  Cao D, Han P. Solutions to critical elliptic equations with muti-sigular inverse square potentials. J Differ  Equ, 2006, 224: 332--372

[11]  Cao D, He X,  Peng S. Positive solutions for some sigular critical growth nonlinear elliptic equations. Nonlinear Anal, 2005, 60: 589--609

[12]  Cao D, Peng S. A note on the sign-changing solutions to elliptic problems with critical Sobolev and Hardy terms. J Differ  Equ, 2003, 193: 424--434

[13] Capozzi A,  Fortunato D, Palmieri G.  An existence result for nonlinear elliptic problems involving critical Sobolev exponent.  Ann Inst H Poincarè Anal Non Linèaire, 1985, 2: 463--470

[14]  Chen J. Existence of solutions for a nonlinear PDE with an inverse square potential. J Differ  Equ, 2003, 195: 497--519

[15]  Chen J. Multiple positive and sign-changing solutions for a sigular Schr\"odinger equation with critical growth. Nonlinear Anal, 2006, 64:  381--400

[16]  Chen J. Multiple positive solutions for a class of nonlinear elliptic equations. J Math Anal Appl, 2004, 295: 341--354

[17]  Chen J. Some further results on a semilinear equation with concave-convex nonlinearity. Nonlinear Anal, 2005, 62: 71--87

[18]  Ekeland I. Ghoussoub N. Selected new aspects of the calculus of variations in the large. Bull Amer Math Soc, 2002, 39:  207--265

[19]  Azorero J P G, Alonso I P. Hardy inequalities and some critical elliptic and parabolic problems. J Differ Equ, 1998, 144: 441--476

[20]  Ghoussoub N, Yuan  C. Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents. Trans Amer Math Soc, 2000, 352:  5703--5743

[21]  Han P. High-energy positive solutions for critical growth Dirichlet problem in noncontractible domains. Nonlinear Anal, 2005, 60: 369--387

[22]  Han  P. The effect of the domain topology on the number of positive solutions of elliptic systems involving critical Sobolev exponents. Houston J Math, 2006, 32: 332--372

[23]  Hsu T S. Multiplicity of positive solutions for critical singular elliptic systems with concave-convex nonlinearities. Adv Nonlinear Stud, 2009, 9: 295--311

[24]  Hsu T S, Lin  H L. Multiple positive solutions for singular elliptic equations with weighted Hardy terms and critical Sobolev-Hardy exponents. Proc Roy Soc Edinburgh Sect A, 2010, 140: 617--633

[25]  Jannelli E.  The role played by space dimension in elliptic critical problems. J Differ  Equ, 1998, 144: 441--476

[26] Kang D, Peng S. Positive solutions for singular critical elliptic problems. Appl Math Lett, 2004, 17: 411--416

[27]  Kang D, Peng S. Existence of solutions for elliptic problems with critical Sobolev-Hardy exponents. Israel J Math, 2004, 143: 281--297

[28]  Liu Z, Han P. Existence of solutions for singular elliptic systems with critical exponents. Nonlinear Anal, 2008, 69: 2968--2983

[29]  Smets D. Nonlinear Schr\"odinger equation with Hardy potential and critical nonlinearities. Trans Amer Math Soc, 2005, 357: 2909--2938

[30] Tarantello G. On nonhomogeneous elliptic involving critical Sobolev exponent.  Ann Inst H Poincar\'e Anal Non Lin\'eaire, 1992, 9: 281--304

[31] Terracini S. On positive entire solutions to a class of equations with critical singular coefficient and critical exponent.  Adv Differ Equ, 1996, 1: 241--264

[32]  Wu T F. On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function. J Math
Anal Appl, 2006, 318:  253--270

Outlines

/