On the Existence of Positive Eigenvalues for Semilinear Elliptic Equation on all of Rd

In this paper, we consider a problem of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\left\{ \begin{gathered} \Delta {\kern 1pt} u\left( x \right) + {\lambda g}\left( x \right)f\left( {u\left( x \right)} \right) = 0\quad x \in \mathbb{R}^d , \hfill \\ \lim _{\left\| x \right\| \to \infty } u\left( x \right) = 0, \hfill \\ \end{gathered} \right.$$ \end{document} where d≥3, f is a positive locally Lipschitz bounded function and g is assumed to change sign. We give some conditions of integral type to get the existence of positive solutions for λ large enough.