Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 41, 2016, 973-1004

INFINITELY MANY SOLUTIONS FOR QUASILINEAR ELLIPTIC EQUATIONS INVOLVING DOUBLE CRITICAL TERMS AND BOUNDARY GEOMETRY

Chunhua Wang and Chang-Lin Xiang

School of Mathematics and Statistics and Hubei Key Laboratory of Mathematical Sciences
Central China Normal University, Wuhan 430079, P.R. China; chunhuawang 'at' mail.ccnu.edu.cn

Yangtze University, School of Information and Mathematics, Jingzhou 434023, P.R. China
and University of Jyväskylä, Department of Mathematics and Statistics
P.,O. Box 35, FI-40014 University of Jyväskylä, Finland; Xiang_math 'at' 126.com

Abstract. In this paper, we study the following problem

–Δ_pu = μ|u|^Np/(N-p-2)u + |u|^{(N-s)p/(N-p)-2}u / |x|^s + a(x)|u|^p-2u in Ω,
u = 0 on ∂Ω,

where 1 < p < N, 0 < s < p, μ ≥ 0 are constants, Δ_p is the p-Laplacian operator, Ω ⊂ R^N is a C² bounded domain with 0 &isin: \barΩ and a ∈ C¹(\barΩ). By an approximation argument, we prove that if N > p² + p, a(0) > 0 and Ω satisfies some geometry conditions if 0 ∈ ∂Ω, for example, all the principle curvatures of ∂Ω at 0 are negative, then the above problem has infinitely many solutions.

2010 Mathematics Subject Classification: Primary 35J60, 35B33.

Key words: Quasilinear elliptic equations, double critical terms, boundary geometry condition, infinitely many solutions, approximation argument.

Reference to this article: C. Wang and C.-L. Xiang: Infinitely many solutions for quasilinear elliptic equations involving double critical terms and boundary geometry. Ann. Acad. Sci. Fenn. Math. 41 (2016), 973-1004.

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doi:10.5186/aasfm.2016.4161