Resonant-Superlinear and Resonant-Sublinear Dirichlet Problems

In this paper, we study elliptic equations in which the reaction (right hand side) exhibits an asymmetric behavior as x→±∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\rightarrow \pm \infty $$\end{document}. More precisely, we assume that we have resonance as x→-∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\rightarrow -\infty $$\end{document}, while as x→+∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\rightarrow +\infty $$\end{document} the equation is superlinear. Using variational tools combined with the theory of critical groups, we prove several multiplicity theorems for nonlinear, nonhomogeneous equations and for semilinear equations (driven by the Laplacian).