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

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 $0\leq s_1 < s_2 < 2$, $2<p< 2^{*}(s_1)$, $q< 2^{*}(s_2)$. 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.