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Multi-bubble nodal solutions to slightly subcritical elliptic problems with Hardy terms in symmetric domains

  • * Corresponding author: Qianqiao Guo

    * Corresponding author: Qianqiao Guo
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  • We consider the slightly subcritical elliptic problem with Hardy term

    $ \left\{ \begin{aligned} - \Delta u-\mu\frac{u}{|x|^2} & = |u|^{2^{\ast}-2- \varepsilon}u &&\quad \rm{in } \Omega\subset{\mathbb{R}}^N, \\\ u & = 0&&\quad \rm{on } \partial \Omega, \end{aligned} \right. $

    where $ 0\in \Omega $ and $ \Omega $ is invariant under the subgroup $ SO(2)\times\{\pm E_{N-2}\}\subset O(N) $; here $ E_n $ denots the $ n\times n $ identity matrix. If $ \mu = \mu_0 \varepsilon^ \alpha $ with $ \mu_0>0 $ fixed and $ \alpha>\frac{N-4}{N-2} $ the existence of nodal solutions that blow up, as $ \varepsilon\to0^+ $, positively at the origin and negatively at a different point in a general bounded domain has been proved in [5]. Solutions with more than two blow-up points have not been found so far. In the present paper we obtain the existence of nodal solutions with a positive blow-up point at the origin and $ k = 2 $ or $ k = 3 $ negative blow-up points placed symmetrically in $ \Omega\cap({\mathbb{R}}^2\times\{0\}) $ around the origin provided a certain function $ f_k:{\mathbb{R}}^+\times{\mathbb{R}}^+\times I\to{\mathbb{R}} $ has stable critical points; here $ I = \{t>0:(t,0,\dots,0)\in \Omega\} $. If $ \Omega = B(0,1)\subset{\mathbb{R}}^N $ is the unit ball centered at the origin we obtain two solutions for $ k = 2 $ and $ N\ge7 $, or $ k = 3 $ and $ N $ large. The result is optimal in the sense that for $ \Omega = B(0,1) $ there cannot exist solutions with a positive blow-up point at the origin and four negative blow-up points placed on the vertices of a square centered at the origin. Surprisingly there do exist solutions on $ \Omega = B(0,1) $ with a positive blow-up point at the origin and four blow-up points on the vertices of a square with alternating positive and negative signs. The results of our paper show that the structure of the set of blow-up solutions of the above problem offers fascinating features and is not well understood.

    Mathematics Subject Classification: 35B44, 35B33, 35J60.


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