On an Ambrosetti-Prodi type problem in ${\mathbb{R}}^N$

In this paper we study results of existence and non-existence of solutions for the following Ambrosetti-Prodi type problem $$\{\begin{array}{l}-\mathrm{\Delta }u=P(x)(g(u)+f(x))in{\mathbb{R}}^N,\\ u\in D^{1,2}({\mathbb{R}}^N),{\lim }_{\mathrm{\mid }x\mathrm{\mid }\to +\mathrm{\infty }}u(x)=0,\end{array}\text{eqno}(P)$$ where $N\ge 3$, $P\in C({\mathbb{R}}^N,{\mathbb{R}}^{+})$, $f\in C({\mathbb{R}}^N)\cap L^{\mathrm{\infty }}({\mathbb{R}}^N)$ and $g\in C^1(\mathbb{R})$. The main tools used are the sub-supersolution method and Leray-Schauder topological degree theory.