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

TL;DR

This paper investigates the existence and non-existence of solutions for a class of Ambrosetti-Prodi type problems in rom ocusing on the conditions under which solutions exist or do not, using topological and sub-supersolution methods.

Contribution

It extends the analysis of Ambrosetti-Prodi problems to rom ocusing on the conditions under which solutions exist or do not, using topological and sub-supersolution methods.

Findings

Established criteria for existence of solutions.

Identified conditions leading to non-existence.

Applied topological degree theory to nonlinear PDEs.

Abstract

In this paper we study results of existence and non-existence of solutions for the following Ambrosetti-Prodi type problem $$ \left\{ \begin{array}{lcl} -\Delta u=P(x)\Big( g(u)+f(x)\Big) \mbox{ in } \mathbb{R}^N,\\ u \in D^{1,2}(\R^N),\ \lim_{|x|\to +\infty}u(x)=0, \end{array} \right. \eqno{(P)} $$ where $N\geq3$, $P\in C(\R^N,\R^+)$, $f\in C(\R^N)\cap L^{\infty}(\R^N)$ and $g\in C^1(\R)$. The main tools used are the sub-supersolution method and Leray-Schauder topological degree theory.