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

Abstract

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