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A Robin boundary problem with Hardy potential and critical nonlinearities

Authors:

Yinbin Deng Lingyu Jin Shuangjie Peng

Institution:

(1) Department of Mathematics, Huazhong Normal University, Wuhan, 430070 Hubei, P. R. China

Abstract:

Let Ω be a bounded domain with a smooth C ² boundary in ℝⁿ (n ≥ 3), 0 ∈ $\bar \Omega$ , and υ denote the unit outward normal to ∂Ω. In this paper, we are concerned with the following class of boundary value problems:

$\left\{ \begin{gathered} - \Delta u - \mu \tfrac{u}{{\left| x \right|^2 }} + \lambda u = \left| u \right|^{2^* - 2} u + \eta \left| u \right|^{p - 2} u, in \Omega , \hfill \\ \tfrac{{\partial u}}{{\partial v}} + \alpha (x)u = 0, on \partial \Omega , \hfill \\ \end{gathered} \right.$

(*)

where 2* = 2n/(n − 2) is the limiting exponent for the embedding of H ¹(Ω) into L ^p(Ω), 2 < p < 2*, $\bar \mu \triangleq \tfrac{{(n - 2)^2 }}{4}$ , η ≥ 0 and λ ∈ ℝ¹ are parameters, and α(x) ∈ C(∂Ω), α(x) ≥ 0. Through a compactness analysis of the functional corresponding to the problem (*), we obtain the existence of positive solutions for this problem under various assumptions on the parameters μ, λ and the fact that 0 ∈ Ω or 0 ∈ ∂Ω. The research was supported by NSFC(10471052, 10571069, 10631030) and the Key Project of Chinese Ministry of Education(107081) and NCET-07-0350.

Keywords:

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