带扰动项的临界奇异双调和方程的非平凡解

Nontrivial Solution for Singular Biharmonic Equation Involving Critical Exponents and Perturbation Terms

研究了一类带扰动项的临界奇异双调和方程Δ2u-μ|xu|s=|u|2*-2u+k(x)|u|q-2u+λu,x∈Ω,u=u/ν=0,x∈Ω,其中ν表示边界Ω的单位外法向量,2*=2N/N-4是嵌入H2(RN)L2*(RN)的临界Sobolev指数,0≤s<4,2<q<2*(s)=2(N-s)/N-4),μ>0,λ>0为参数。利用Sobolev嵌入的最佳达到函数和精确的能量估计,运用山路引理得到了这类双调和方程非平凡解的存在性。

In this paper,we study the singular critical biharmonic equation with perturbation terms Δ2u-μ u｜x｜s=｜u｜2＊-2u＋k（x）｜u｜q-2u＋λu,x ∈Ω,u=ν/u=0,x∈Ω,where vdenotes the unit outward normal vector to boundary Ω,2＊=2N/N-4 is the critical Sobolev exponent for the embedding H2（RN）L2＊（RN）,0 ≤s4,2q 2＊（s）=2（N-s）/N-4,μ0,λ0 are parameters.By the best attained function of Sobolev embedding and delicate energy estimates.With the mountain pass theorem,the existence of nontrivial solution of the biharmonic equation is obtained.