Resonant-Superlinear and Resonant-Sublinear Dirichlet Problems

被引：0
作者：
Zhenhai Liu
Nikolaos S. Papageorgiou
机构：
[1] Yulin Normal University,Center for Applied Mathematics of Guangxi
[2] Guangxi Minzu University,Guangxi Key Laboratory of Universities Optimization Control and Engineering Calculation
[3] National Technical University,Department of Mathematics
来源：
Bulletin of the Malaysian Mathematical Sciences Society | 2024年 / 47卷
关键词：
Asymmetric reaction; Regularity theory; Maximum principle; Resonance; Critical groups; 35J20; 35J60; 58E05;
D O I：
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摘要：
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).
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