MULTIPLE SOLUTIONS FOR THE p&q-LAPLACIAN PROBLEM WITH CRITICAL EXPONENT

doi:10.1016/S0252-9602(09)60077-1

Abstract

Abstract:

In this paper, we study the existence of multiple solutions for the following nonlinear elliptic problem of p&q-Laplacian type involving the critical Sobolev exponent:

−△_pu − △_qu = |u|^p*−2u + μ|u|^r−2u in Ω,
u|∂Ω= 0,

where Ω ⊂ R^N is a bounded domain, N > p, p* = N_p /N−p is the critical Sobolev exponent and μ > 0. We prove that if 1 < r < q < p < N, then there is a μ₀ > 0, such that for any μ ∈ (0, μ₀), the above mentioned problem possesses infinitely many weak solutions. Our result generalizes a similar result in [8] for p-Laplacian type problem.

Key words: p&q-Laplacian, multiplicity of solutions, critical exponent

CLC Number:

35J60

LI Gong-Bao, ZHANG Guo. MULTIPLE SOLUTIONS FOR THE p&q-LAPLACIAN PROBLEM WITH CRITICAL EXPONENT[J].Acta mathematica scientia,Series B, 2009, 29(4): 903-918.

Trendmd

References

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

[2] Ambrosetti A, Rabinowitz P H. Dual variational methods in critical point thoery and application. J Func
Anal, 1973, 14: 349–381

[3] Brezis H. Nonlinear equation involving the critical Sobolev exponent-survey and perspectives//Crandall
M C, et al, ed. Directions in Partial Differential Equations. New York: Academic Press Inc, 1987: 17–36

[4] Brezis H, Lieb E H. A relation between pointwise convergent of functions and convergent of functional.
Proc Amer Math Soc, 1983, 88: 486–490

[5] Brezis H, Nirenberg L. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent.
Comm Pure Appl Math, 1983, 36: 437–477

[6] Benci V, Micheletti A M. Visetti D. An eigenvalue problem for a quasilinear elliptic field equation. J
Differential Equations, 2002, 184(2): 299–320

[7] Cherfils L, Il’yasov Y. On the stationary solutions of generalized reaction diffusion equation with p&q-
Laplacian. Comm Pure Appl Anal, 2005, 4(1): 9–22

[8] Azorero Garcia J, Aloson Peral I. Multiplicity of solutions for elliptic problems with critical exponent or
with a nonsymmetric term. Trans Amer Math Soc, 1991, 323: 877–895

[9] Ladyzenskaya O, Uralsteva N. Linear and Quasilinear Elliptic Partial Differential Equations. New York:
Academic Press, 1968

[10] Li G B. The existence of nontrivial solutions of quasilinear elliptic PDE of variational type [D]. Wuhan:
Wuhan Univ (in Chinese), 1987

[11] Li G B. The existence of a nontrivial solution to the p&q-Laplacian problem with nonlinearity asymptotic
to u^p−1 at infinity in RN. Nonlinear Anal, 2008, 68: 1100–1119

[12] Li G B, Martio O. Stability in obstacle problem. Math Scand, 1994, 75: 87–100

[13] Lions P L. The concentraction-compactness principle in the calculus of virations. The limit case, Part 1.
Rev Mat Iberoamericana, 1985, 1: 145–201

[14] Lions P L. The concentraction-compactness principle in the calculus of virations. The limit case, Part 2.
Rev Mat Iberoamericana, 1985, 2: 45–121

[15] Rabinowitz P H. Minimax methods in critical points thoery with application to differential equations. CBMS Regional Conf Ser in Math, Vol 65. Providence, RI: Amer Math Soc, 1986

[16] Solimini S. On the existence of infinitely many radial solutions for some elliptic problems. Revista Mat
Aplicadas, 1987, 9: 75–86

[17] Zhong X. Multiplicity of solutions for quasilinear elliptic equations with critical exponet on R^N [D].Wuhan:
Wuhan Institute of Mathematical Sciences, Chinese Academy of Sciences, 1995

[18] Zhu X P. Nontrivial solution of quasilinear elliptic equations involving critical Sobolev exponent. Sciences
Sinica Ser A, 1988, 31: 1166–1181