Ground state solution for nonlocal scalar field equations involving an integro-differential operator

This paper is concerned with nonlocal scalar field equations involving an integro-differential operator. We investigate the existence of solutions for the problem -LKu=g(u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} -\mathcal {L}_{K}u=g(u) \end{aligned}$$\end{document}in RN\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}, where -LK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\mathcal {L}_{K}$$\end{document} is an integro-differential operator. Under appropriate hypotheses, we prove that this equation has a ground state solution.