On Positive Solutions of Elliptic Equations with Oscillating Nonlinearity in $\mathbb {R}^N$

Abstract

In this paper, we study results of existence and multiplicity of positive solutions for the following semilinear problem

$$\begin{aligned} \left\{ \begin{array}{lcl} -\Delta u= \lambda P(x)f(u)\, \text{ in } \, \mathbb {R}^N\\ \lim \limits _{|x|\rightarrow \infty }u(x)=0, \end{array} \right. \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad {(P)} \end{aligned}$$

where $P\in C(\mathbb {R}^N,\mathbb {R})$ and $f\in C([0,\infty ),\mathbb {R})$ is an oscillating nonlinearity satisfying a sort of area condition. The main tools used are variational methods and sub-supersolution method.