The Li-Lin's open problem on $\mathbb{R}^N$

TL;DR

This work extends the Li–Lin open problem on the existence of positive solutions to a Hardy–Sobolev type equation from bounded domains to the whole space $\mathbb{R}^N$, highlighting a fundamental contrast between the two settings. The authors analyze $-\Delta u+u = -\lambda|x|^{-s_{1}}|u|^{p-2}u + |x|^{-s_{2}}|u|^{q-2}u$ with $0\le s_{1}<s_{2}<2$ and establish nonexistence when $q=2^{*}(s_{2})$ for all $\lambda>0$ and existence for subcritical $q<2^{*}(s_{2})$ under a parametric relation between $p$ and $q$. The Nehari-manifold framework is developed but the energy functional is unbounded below on the manifold, so the authors identify a local minimizer on $M^+$ and obtain a positive solution via Ekeland’s variational principle and radial symmetry arguments. The results reveal distinct phenomena in $\mathbb{R}^N$ compared to bounded domains and advance understanding of Li–Lin-type problems in unbounded settings.

Abstract

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.

Theorems & Definitions (12)

• Theorem 1.1
• Theorem 1.3
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Proposition 2.4
• proof