Weak solutions of semilinear elliptic equation involving Dirac mass

Abstract

In this paper, we study the elliptic problem with Dirac mass

where $N>2$, $p>0$, $k>0$, ${\delta }_0$ is the Dirac mass at the origin and the potential $V$ is locally Lipchitz continuous in ${\mathbb{R}}^N\setminus \{0\}$, with non-empty support and satisfying

with $a_0<N$, $a_0<a_{\infty }$ and ${\sigma }_1>0$. We obtain two positive solutions of (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 via Mountain Pass Theorem.