Acta mathematica scientia, Series B >
p-LAPLACE EQUATIONS WITH MULTIPLE CRITICAL EXPONENTS AND SINGULAR CYLINDRICAL POTENTIAL
Received date: 2012-04-19
Revised date: 2012-06-04
Online published: 2013-07-20
Supported by
Supported by the National Science Foundation of China (11071245 and 11101418).
In this paper, we deal with the following problem:
{−Δpu − λ|y|−p|u|p−2u = |y|−s|u|p*(s)−2u + |u|p*−2u in RN, y ≠ 0,
u ≥ 0,
where u(x) = u(y, z) : Rm × RN−m −→ R, N ≥ 3, 2 < m < N, 1 < p < m, λ <( (m−p)/p )p and 0 < s < p, p*(s) = p(N−s)/N−p , p* = pN/N−p . By variational method, we prove the existence of a nontrivial weak solution when 0 < λ <( (m−p/ p ))p and the existence of a cylindrical weak solution when λ < 0.
Key words： p-Laplace equation; cylindrical potential; critical exponents
SUN Xiao-Mei . p-LAPLACE EQUATIONS WITH MULTIPLE CRITICAL EXPONENTS AND SINGULAR CYLINDRICAL POTENTIAL[J]. Acta mathematica scientia, Series B, 2013 , 33(4) : 1099 -1112 . DOI: 10.1016/S0252-9602(13)60066-1
[1] Alves C O. Existence of positive solutions for a problem with lack of compactness involving the p-Laplacian. Nonlinear Anal, 2002, 51: 1187–1206
[2] Abdellaoui B, Felli V, Peral I. Existence and nonexistence results for quasilinear elliptic equations involving the p-laplacian. Boll Unione Mat Ital Sez B Artic Ric Mat, 2006, 9: 445–484
[3] Badiale M, Bergio V, Rolando S. A nonlinear elliptic equation with singular potential and applications to nonlinear field equations. J Eur Math Soc, 2007, 9: 355–381
[4] Badiale M, Guida M, Rolando S. Elliptic equations with decaying cylindrical potentials and power-type nonlinearities. Adv Diff Equ, 2007, 12: 1321–1362
[5] Badiale M, Tarantello G. A Sobolev-Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics. Arch Ration Mech Anal, 2002, 163: 252–293
[6] Bhakta M, Sandeep K. Hardy-Sobolev-Maz’ya type equations in bounded domains. J Differ Equ, 2009, 247: 119–139
[7] Br´ezis H, Lieb E. A relation between pointwise convergence of functionals and convergence of functionals. Proc Amer Math Soc, 1983, 28: 486–490
[8] Br´ezis H, Nirenberg L. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent. Comm Pure Appl Math, 1983, 36: 437–477
[9] Cao D M, Han P G. Solutions for semilinear elliptic equations with critical exponents and Hardy potential. J Differ Equ, 2004, 205: 521–537
[10] Cao D M, Peng S J, Yan S S. Infinitely many solutions for p-Laplacian equation involving critical Sobolev growth. J Funct Anal, 2012, 262: 2861–2902
[11] Chen S W, Lin L S. Results on entire solutions for a degenerate critical elliptic equation with anisotropic coefficients. Sci China Math, 2011, 54: 221–242
[12] Deng Y B, Xie H Z. Multiple stationary solutions of Euler-Poisson equations for nonisentropic gaseous stars. Acta Math Sci, 2010, 30B(6): 2077–2088
[13] Filippucci R, Pucci P, Robert F. On a p-Laplace equation with multiple critical nonlinearities. J Math Pure Appl, 2009, 91: 156–177
[14] Gazzini M, Ambrosetti A, Musina R. Nonlinear elliptic problems related to some integral inequalities.http://digitallibrary.sissa.it/retrieve/4320/PhDThesisGazzini.pdf.
[15] Gazzini M, Mussina R. On a Sobolev type inequality related to the weighted p-Laplace operator. J Math Anal Appl, 2009, 352: 99–111
[16] Kang D S, Li G Q. On the elliptic problems involving multi-singular inverse square potentials and multicritical Sobolev-Hardy exponents. Nonlinear Analysis, 2007, 66: 1806–1816
[17] Li Y Y, Guo Q Q, Niu P C. Global compactness results for quasilinear elliptic problems with combined critical Sobolev-Hardy terms. Nonlinear Analysis, 2011, 74: 1445–1464
[18] Lions P L. The concentration-compactness principle in the Calculus of Variations. The limit case. Part I. Rev Mater Iberoamericano, 1985, 1: 145–201
[19] Lions P L. The concentration-compactness principle in the Calculus of Variations. The limit case. Part II. Rev Mater Iberoamericano, 1985, 1: 45–121
[20] Mercuri C, Willem M. A global compactness result for the p-Laplacian involving critical nonlinearities. Discrete Contin Dyn Syst, 2010, 28: 469–493
[21] Musina R. Ground state solutions of a critical problem involving cylindrical weights. Nonlinear Anal, 2008, 68: 3972–3986
[22] Saintier N. Asymptotic estimates and blow-up theory for critical equations involving the p-Laplacian. Calc Var, 2006, 25: 299–331
[23] Smets D. Nonlinear Schr¨odinger equations with Hardy potential and critical nonlinearities. Trans Amer Math Soc, 2005, 357: 2909–2938
[24] Solimini S. A note on compactness-type properties with respect to Lorentz norms of bounded subsets of a Sobolev space. Ann Inst Henry Poincar´e-Analyse Nonlin´eaire, 1995, 12: 319–337
[25] Struwe M. Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems. 3rd ed. Berlin: Springer-Verlag, 2000
[26] Struwe M. A global compactness result for elliptic boundary value problems involving limiting nonlinearities. Math Z, 1984, 187: 511–517
[27] Terracini S. On positive entire solutions to a class of equations with a singular coefficient and critical exponent. Adv Differ Equ, 1996, 2: 241–264
[28] Tertikas T, Tintarev K. On existence of minimizers for the Hardy-Sobolev-Maz’ja inequality. Ann Mat Purae Appl, 2007, 186: 645–662
[29] Xuan B J. Multiple solutions to p-Laplacian equation with singularity and cylindrical symmetry. Nonlinear Analysis, 2003, 55: 217–232
[30] Xuan B J, Wang J. Existence of a nontrivial weak solution to quasilinear elliptic equations with singular weights and multiple critical exponents. Nonlinear Analysis, 2010, 72: 3649–3658
/
| 〈 |
|
〉 |