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A global compact result for a semilinear elliptic problem with Hardy potential and critical non-linearities on ℝN

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Abstract

In this paper, we are concerned with the following class of elliptic problems:

$$ \left\{ \begin{gathered} - \Delta u - \mu \frac{u} {{|x|^2 }} + a(x)u = |u|^{2* - 2} u + k(x)|u|^{q - 2} u, \hfill \\ u \in H^1 (\mathbb{R}^N ), \hfill \\ \end{gathered} \right. $$

where 2* = 2N/(N - 2) is the critical Sobolev exponent, 2 < q < 2*, \( 0 \leqslant \mu < \bar \mu \triangleq \frac{{(N - 2)^2 }} {4} \), a(x), k(x) ∈ C(ℝN). Through a compactness analysis of the functional corresponding to the problems (*), we obtain the existence of positive solutions for this problem under certain assumptions on a(x) and k(x).

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Authors and Affiliations

  1. College of Science, South China Agriculture University, Guangzhou, 510642, China

    LingYu Jin

  2. Department of Mathematics, Huazhong Normal University, Wuhan, 430079, China

    YinBin Deng

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  1. LingYu Jin
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  2. YinBin Deng
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Correspondence to YinBin Deng.

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Jin, L., Deng, Y. A global compact result for a semilinear elliptic problem with Hardy potential and critical non-linearities on ℝN . Sci. China Math. 53, 385–400 (2010). https://doi.org/10.1007/s11425-009-0075-x

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  • DOI: https://doi.org/10.1007/s11425-009-0075-x

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