Acta mathematica scientia, Series B >
CRITICAL EXPONENTS AND CRITICAL DIMENSIONS FOR NONLINEAR ELLIPTIC PROBLEMS WITH SINGULAR COEFFICIENTS
Received date: 2012-10-09
Revised date: 2013-11-22
Online published: 2014-09-20
Supported by
This work was supported by the National Natural Science Foundation of China (11326139, 11326145), Tian Yuan Foundation (KJLD12067), and Hubei Provincial Department of Education (Q20122504).
Let B1 ⊂ RN be a unit ball centered at the origin. The main purpose of this paper is to discuss the critical dimension phenomenon for radial solutions of the following quasilinear elliptic problem involving critical Sobolev exponent and singular coefficients:
{−div(|∇u|p−2∇u) = |x|s|u|p*(s)−2u + λ|x|t|u|p−2u, x ∈ B1,
u|∂B1 = 0,
where t, s > −p, 2 ≤ p < N, p*(s) = (N+s)p/N−p and λ is a real parameter. We show particularly that the above problem exists infinitely many radial solutions if the space dimension N >p(p − 1)t + p(p2 − p + 1) and λ ∈ (0, λ1,t), where λ1, t is the first eigenvalue of −Δp with the Dirichlet boundary condition. Meanwhile, the nonexistence of sign-changing radial solutions is proved if the space dimension N ≤(ps+p)min{1, p+t/p+s }+p2/p−(p−1)min{1, p+t/p+s } and λ > 0 is small.
WANG Li , WANG Ji-Xiu . CRITICAL EXPONENTS AND CRITICAL DIMENSIONS FOR NONLINEAR ELLIPTIC PROBLEMS WITH SINGULAR COEFFICIENTS[J]. Acta mathematica scientia, Series B, 2014 , 34(5) : 1603 -1618 . DOI: 10.1016/S0252-9602(14)60107-7
[1] Br´ezis H, Nirenberg L. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm Pure Appl Math, 1983, 36: 437–477
[2] Pucci P, Serrin J. Critical exponents and critical dimensionss for polyharimonic operators. J Math Pures Appl, 1990, 69: 55–83
[3] Jones C. Radial solutions of a semilinear elliptic equation at a critical exponent. Arch Rational Mech Anal, 1988, 104: 251–270
[4] Atkinson F V, Br´ezis H, Peletier L A. Nodal solutions of elliptic equations with critical Sobolev exponents. J Differ Equ, 1990, 85: 151–170
[5] Adimurthi, Yadava S L. Elementary proof of the nonexistence of nodal solutions for the semilinear elliptic equations with critical Sobolev exponent. Nonlinear Anal, 1990, 14: 785–787
[6] Bae S, Pahk D H. Nonexistence of nodal solutions of nonlinear elliptic equations. Nonlinear Anal, 2001, 46: 1111–1122
[7] Cao D M, Peng S J. A note of the sign-changing solutions to elliptic problem with critical Sobolev and Hardy terms. J Differ Equ, 2003, 193: 424–434
[8] Castro A, Kurepa A. Radially symmetric solutions to a Dirichlet problem involving critical exponents. Trans Amer Math Soc, 1994, 343: 907–926
[9] Wang X, Wu S. Some notes on radial solutions to the equation Δu+|u|p−1u+Δu = 0. Chinese Ann Math Ser A (in Chinese), 1991, 12: 566–572
[10] Deng Y B, Wang J X. Nonexistence of radical node solutions for elliptic problems with critical Sobolev exponents. Nonlinear Anal, 2009, 71: 172–178
[11] Egnell H. Elliptic boundary value problems with singular coefficients and critical nonlinearities. Indiana Univ Math J, 1989, 38: 235–251
[12] Filippucci R, Ricci R G, Pucci P. Non-existence of nodal and one-signed solutions for nonlinear variational equations. Arch Rational Mech Anal, 1994, 127: 255–280
[13] Ghoussoub N, Yuan C. Multiple solutions for quasilinear PDEs involving the critical Sobolev and Hardy exponents. Trans Amer Math Soc, 2000, 352: 5703–5743
[14] Kang D S, Peng S J. Existence of solutions for elliptic equations with critical Sobolev-Hardy exponents. Nonlinear Anal, 2004, 56: 1151–1164
[15] Wang Y J, Shen Y T. Multiple and sign-changing solutions for nonlinear elliptic equation with critical potential and critical parameter. Acta Math Sci, 2010, 30B(1): 113–124
[16] Cerami G, Solimini S, Struwe M. Some existence results for superlinear elliptic boundary value problem involving critical exponents. J Funct Anal, 986, 69: 289–306
[17] Chen W. Infinitely many solutions for a nonlinear elliptic equation involving critical Sobolev exponents. Acta Math Sci, 1991, 11: 128–135
[18] Tarantello G. Nodal solutions of semilinear elliptic equations with critical exponent. Differ Int Equ, 1982, 5: 25–42
[19] Filippucci R, Ricci R G, Pucci P. Non-existence of nodal and one-signed solutions for nonlinear variational equations. Arch Rational Mech Anal, 1994, 127(3): 255–280
[20] Filippucci R, Ricci R G. Non-existence of nodal and one-signed solutions for nonlinear variational equations with special symmetries. Arch Rational Mech Anal, 1994, 127(3): 281–295
[21] Talenti G. Best constant in Sobolev inequality. Annali Di Mat, 1976, 110: 353–372
[22] Bae S, Choi H O, Pahk D H. Existence of nodal solutions of nonlinear ellptic equations. Proc Roy Soc Edinburgh: Sec A, 2007, 137: 1135–1155
[23] Lions P L. The concentration-compactness principle in the calculus of variations. The limit case I. Rev Mat
Iberoamericana, 1985, 1: 145–201
[24] Deng Y B, Guo Z H, Wang G S. Nodal solutions for p-Laplace equations with critical growth. Nonlinear Anal TMA, 2003, 54: 1121–1151
[25] Chou K, Chu C. On the best constant for a weightened Sobolev-Hardy inequality. J London Math Soc, 1993, 48(2): 137–151
[26] Zhu X P, Yang J F. Regularity for quasilinear elliptic equations in involving critical sobolev exponent. System Sci Math, 1989, 9: 47–52
[27] Vazquez J L. A strong maximum principle for some quasilinear elliptic equations. Appl Math Optim, 1984, 12: 191–202
[28] Cao D M, Zhu X P. On the existence and nodal character of solutions of semilinear elliptic equations. Acta Math Sci, 1988, 8: 345–359
/
| 〈 |
|
〉 |