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Large number of bubble solutions for the equation \(\Delta u + K(y)u^{\tfrac{{N + 2}} {{N - 2}} \pm \varepsilon } \) on ℝN

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Abstract

This paper is concerned with the following nonlinear elliptic equation:

$\left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\Delta u + K(y)u^{\tfrac{{N + 2}} {{N - 2}} \pm \varepsilon } = 0,} & {u > 0,} & {y \in \mathbb{R}^N ,} \\ \end{array} } \\ {u \in D^{1,2} (\mathbb{R}^N ),} \\ \end{array} } \right.$

where ɛ > 0, N ⩾ 5, K(y) is positive and radially symmetric. We show that, under some local conditions on K(y), this problem has large number of bubble solution if ɛ is small enough. Moreover, for each m ∈ [2, N−2), there exists solutions whose functional energy is in the order of \(\varepsilon - \tfrac{{N - 2 - m}} {{(N - 2)^2 }} \).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Henan University, Kaifeng, 475004, China

    ZhongYuan Liu

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  1. ZhongYuan Liu
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Liu, Z. Large number of bubble solutions for the equation \(\Delta u + K(y)u^{\tfrac{{N + 2}} {{N - 2}} \pm \varepsilon } \) on ℝN . Sci. China Math. 59, 459–478 (2016). https://doi.org/10.1007/s11425-015-5053-x

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