Eigenvalue, bifurcation and convex solutions for Monge-Ampère equations
DOI:
https://doi.org/10.12775/TMNA.2015.041Keywords
Eigenvalue, bifurcation, convex solution, Monge--Amp\`{e}re equationAbstract
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.Downloads
Published
2015-09-01
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1.
DAI, Guowei and MA, Ruyun. Eigenvalue, bifurcation and convex solutions for Monge-Ampère equations. Topological Methods in Nonlinear Analysis. Online. 1 September 2015. Vol. 46, no. 1, pp. 135 - 163. [Accessed 29 March 2024]. DOI 10.12775/TMNA.2015.041.
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