Description

• Title:
  Eigenvalue, bifurcation and convex solutions for Monge-Ampère equations
• Creator:
  Dai, Guowei;
  Ma, Ruyun
• Description:
  In this paper we study the following eigenvalue boundary value problem for Monge-Amp\`{e}re equations \det(D^2u)=\lambda^N f(-u) \text{in } \Omega, u=0 \text{on } \partial \Omega. We establish global bifurcation results for the problem with $f(u)=u^N+g(u)$ and $\Omega$ being the unit ball of $\mathbb{R}^N$. More precisely, under some natural hypotheses on the perturbation function $g\colon[0,+\infty)\rightarrow[0,+\infty)$, we show that $(\lambda_1,0)$ is a bifurcation point of the problem and there exists an unbounded continuum of convex solutions, where $\lambda_1$ is the first eigenvalue of the problem with $f(u)=u^N$. As the applications of the above results, we consider with determining interval of $\lambda$, in which there exist convex solutions for this problem in unit ball. Moreover, we also get some results about the existence and nonexistence of convex solutions for this problem on general domain by domain comparison method.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2015 Topological Methods in Nonlinear Analysis
• Date:
  2015.09.01
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 46, No 1 (September 2015); 135 - 163;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2015.041/7204
• Subject:
  Eigenvalue;
  bifurcation;
  convex solution;
  Monge--Amp\`{e}re equation;
  34C23;
  34D23;
  35J60
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2015.041
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other