Weak solutions of semilinear elliptic equation involving Dirac mass

Abstract

In this paper, we study the following elliptic problem with Dirac mass \begin{equation}\label{eq 0.1} -Δu=Vu^p+k δ_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.

23 pages