The Li-Lin's open problem on $\mathbb{R}^N$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916722683936768 |
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| author | Tang, Zhi-Yun Tang, Xianhua |
| author_facet | Tang, Zhi-Yun Tang, Xianhua |
| contents | In 2012, Y.Y. Li and C.-S. Lin (Arch. Ration. Mech. Anal., 203(3): 943-968) posed an open problem concerning the existence of positive solutions to the elliptic equation $$
\begin{cases}
-Δu = -λ|x|^{-s_1}|u|^{p-2}u + |x|^{-s_2}|u|^{q-2}u & \text{in } Ω,
u = 0 & \text{on } \partial Ω,
\end{cases} $$ for $λ> 0$, $p > q = 2^*(s_2)$, $0 \leq s_1 < s_2 < 2$, and $2^*(s) = \frac{2(N-s)}{N-2}$ denotes the Hardy-Sobolev critical exponent, initially studied in bounded domains $Ω\subset \mathbb{R}^N$, $N \geq 3$. Currently, research on this open problem remains limited, and a complete resolution is still far from being achieved.
Motivated by the need to address this open problem in more general settings, we extend our investigation to the entire space $\mathbb{R}^N$, focusing on the equation $$ -Δu + u = -λ|x|^{-s_1}|u|^{p-2}u + |x|^{-s_2}|u|^{q-2}u \quad \text{in } \mathbb{R}^N. $$
Our analysis reveals stark contrasts between bounded and unbounded domains: in $\mathbb{R}^N$, the equation admits no solution when $q = 2^*(s_2)$ for any $λ> 0$, whereas a positive solution exists when $q < 2^*(s_2)$.
To establish these results, we employ the Nehari manifold method; however, the functional's unboundedness from below on the manifold causes standard global minimization techniques to be inapplicable. Instead, we characterize a local minimizer of the energy functional on the Nehari manifold, overcoming the challenge posed by the lack of a global minimizer. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_03613 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Li-Lin's open problem on $\mathbb{R}^N$ Tang, Zhi-Yun Tang, Xianhua Analysis of PDEs Functional Analysis In 2012, Y.Y. Li and C.-S. Lin (Arch. Ration. Mech. Anal., 203(3): 943-968) posed an open problem concerning the existence of positive solutions to the elliptic equation $$ \begin{cases} -Δu = -λ|x|^{-s_1}|u|^{p-2}u + |x|^{-s_2}|u|^{q-2}u & \text{in } Ω, u = 0 & \text{on } \partial Ω, \end{cases} $$ for $λ> 0$, $p > q = 2^*(s_2)$, $0 \leq s_1 < s_2 < 2$, and $2^*(s) = \frac{2(N-s)}{N-2}$ denotes the Hardy-Sobolev critical exponent, initially studied in bounded domains $Ω\subset \mathbb{R}^N$, $N \geq 3$. Currently, research on this open problem remains limited, and a complete resolution is still far from being achieved. Motivated by the need to address this open problem in more general settings, we extend our investigation to the entire space $\mathbb{R}^N$, focusing on the equation $$ -Δu + u = -λ|x|^{-s_1}|u|^{p-2}u + |x|^{-s_2}|u|^{q-2}u \quad \text{in } \mathbb{R}^N. $$ Our analysis reveals stark contrasts between bounded and unbounded domains: in $\mathbb{R}^N$, the equation admits no solution when $q = 2^*(s_2)$ for any $λ> 0$, whereas a positive solution exists when $q < 2^*(s_2)$. To establish these results, we employ the Nehari manifold method; however, the functional's unboundedness from below on the manifold causes standard global minimization techniques to be inapplicable. Instead, we characterize a local minimizer of the energy functional on the Nehari manifold, overcoming the challenge posed by the lack of a global minimizer. |
| title | The Li-Lin's open problem on $\mathbb{R}^N$ |
| topic | Analysis of PDEs Functional Analysis |
| url | https://arxiv.org/abs/2505.03613 |