一类超临界椭圆方程正解的存在性

On the Existence of Positive Solutions of Elliptic Equations with Supercritical Growth

中文关键词:  半线性椭圆方程  分歧  超临界Sobolev指数

英文关键词:semilinear elliptic equations  bifurcations  supercritical Sobolev exponents.

中文摘要:

      本文讨论了球上半线性椭圆Ｄｉｒｉｃｈｌｅｔ问题Δｕ＋λｕ^ｑ＋ｕ^p＝０正解的存在性，其中，λ∈Ｒ，０〈ｑ〈１，ｐ〉ｐ_ｃ≡(Ｎ＋２)/(Ｎ－２)（Ｎ〉２）．在条件Ｎ≤６或Ｎ〉６，ｐ〈ｐ_Ｎ≡（Ｎ＋１－(２Ｎ－３)^1/2）／（Ｎ－３－(２Ｎ－３)^1/2）下，证明了存在唯一的λ₀，λ₀〉０，当λ＝λ₀时，有唯一的径向奇异解及无穷多个正解。

英文摘要:

      This paper deals with the existence of the positive solutions of the semilinear elliptic Dirichlet problem Δｕ＋λｕ^ｑ＋ｕ^p＝０ on a ball where ０〈ｑ〈１，ｐ〉ｐ_ｃ≡(Ｎ＋２)/(Ｎ－２)（Ｎ〉２） and λ∈R . Under the condition N<6 or N〉6, p〈p_N , we prove that there exists a unique constant λ₀〉0 , such that for λ=λ₀ , there exists a unique radial singular solution and infinitely many of solutions.