Nonlinear perturbations of a periodic elliptic problem with discontinuous nonlinearity in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}^{N}}$$\end{document}

Using variational methods, we establish existence of positive solutions for a class of elliptic problems like \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\Delta{u}+V(x)u=H(u-\beta)f(u)\,\,\,\, {\rm in}\,\,\,\mathbb{R}^{N},$$\end{document}where β > 0, V is a positive, continuous perturbations of a periodic function, H is the Heaviside function and f is a continuous function with subcritical growth.