On the Existence of Positive Solutions of Nonlinear Elliptic Equations

We study the existence of positive solutions of the nonlinear elliptic problem \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\frac{1}{2}\Delta u - f(u) \cdot \mu + g(u) \cdot \sigma = 0$$ \end{document} in D with u=0 on ∂D, where μ and σ are two Randon's measures belonging to a Kato subclass and D is an unbounded smouth domain in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{R}$$ \end{document}d(d≥3). When g is superlinear at 0 and 0≤f(t)≤t for t∈(0,b), then probabilistic methods and fixed point argument are used to prove the existence of infinitely many bounded continuous solutions of this problem.