Weak solutions of semilinear elliptic equation involving Dirac mass

TL;DR

This paper investigates the existence of weak solutions to a semilinear elliptic equation with a Dirac mass, establishing two positive solutions under specific parameter conditions using variational methods.

Contribution

The paper introduces new existence results for positive solutions to elliptic equations with singular sources, employing variational techniques and the Mountain Pass theorem.

Findings

Existence of a minimal positive solution.

Construction of a second solution via Mountain Pass theorem.

Conditions on parameters ensuring solution existence.

Abstract

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.