A positive solution of the elliptic equation on a starshaped domain with boundary singularities
Zhi-Yun Tang, Xianhua Tang
TL;DR
The paper tackles a weighted elliptic equation with boundary singularities on a bounded star-shaped domain, seeking positive solutions in a subcritical regime. It employs a variational approach on the Nehari manifold, establishing a local minimum on $M^+$ under the inequality $p>q>\frac{2-s_2}{2-s_1}p+\frac{2s_2-2s_1}{2-s_1}$, with the energy functional $I(u)$ defined on $H^1_0(\Omega)$. It also analyzes the asymptotic behavior of the positive solution and identifies a new class of blow-up points located on the boundary via blow-up analysis, distinguishing them from interior blow-up scenarios. These results provide subcritical approximations to Li-Lin's open problem and reveal boundary-dominated blow-up phenomena in this weighted elliptic setting.
Abstract
We consider the elliptic equation with boundary singularities \begin{equation} \begin{cases} -Δu=-λ|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 $0\leq s_1 < s_2 < 2$, $2<p< 2^{*}(s_1)$, $q< 2^{*}(s_2)$. Which is the subcritical approximations of the Li-Lin's open problem proposed by Li and Lin (Arch Ration Mech Anal 203(3): 943-968, 2012). We find a positive solution which is a local minimum point of the energy functional on the Nehari manifold when $p>q>\frac{2-s_2}{2-s_1}p+\frac{2s_2-2s_1}{2-s_1}$. 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.