In this paper, we study the following elliptic problem with Dirac mass \begin{ equation}\label{ eq 0.1} -\Delta u=Vu^p+k \delta_0\quad { \rm in}\quad \mathbb{ R}^N, \qquad \lim_{ |x|\to+\infty}u(x)=0, \end{ equation} where $N>2$, $p>0$, $k>0$, $\delta_0$ is Dirac mass at the origin, the function $V$ is a locally Lipchitz continuous in $\mathbb{ R}^N\setminus\{ 0\}$ satisfying $$ V(x)\le \frac{ c_1}{ |x|^{ a_0}(1+|x|^{ a_\infty-a_0})} $$ with $a_0<N,\ a_\infty>a_0 $ and $c_1>0$. We obtain two positive solutions of (\ref{ eq 0.1}) with additional conditions for parameters on $a_\infty, a_0$, $p$ and $k$. The first solution is a minimal positive solution and the second solution is constructed by Mountain Pass theorem.
Categories: math.AP