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arXiv · 2504.16280

A positive solution of the elliptic equation on a starshaped domain with boundary singularities

Zhi-Yun Tang·Xianhua Tang
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Abstract

We consider the elliptic equation with boundary singularities \begin{equation} \begin{cases} -\Delta u=-\lambda |x|^{-s_{1}}|u|^{p-2}u+|x|^{-s_{2}}|u|^{q-2}u &\text { in } \varOmega , u(x)=0 &\text { on } \partial \varOmega , \end{cases} \end{equation} where $0\leq s_1 < s_2 < 2$, $2 q>\frac{2-s_2}{2-s_1}p+\frac{2s_2-2s_1}{2-s_1}$. We also discuss the asymptotic behavior of the positive solution and find a new class of blow-up points by blowing up analysis. These blow-up points are on the boundary of the domain, which are not similar with the usual.

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Zhi-Yun Tang, Xianhua Tang. 2025-04-22. A positive solution of the elliptic equation on a starshaped domain with boundary singularities. https://arxiv.org/abs/2504.16280

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