Stable solutions for the bilaplacian with exponential nonlinearity

Details Stable solutions for the bilaplacian with exponential nonlinearity Stable solutions for the bilaplacian with exponential nonlinearity

2008-01-16

arxiv.org/abs/0801.2445v1

Mathematics

Analysis of PDEs

Scientific paper

Let $\lambda^*>0$ denote the largest possible value of $\lambda$ such that \begin{align*} \left\{\begin{aligned} \Delta^2 u & = \la e^u && \text{in $B $} u &= \pd{u}{n} = 0 && \text{on $ \pa B $} \end{aligned} \right. \end{align*} has a solution, where $B$ is the unit ball in $\R^N$ and $n$ is the exterior unit normal vector. We show that for $\lambda=\lambda^*$ this problem possesses a unique {\em weak} solution $u^*$. We prove that $u^*$ is smooth if $N\le 12$ and singular when $N\ge 13$, in which case $ u^*(r) = - 4 \log r + \log (8(N-2)(N-4) / \lambda^*) + o(1)$ as $r\to 0$. We also consider the problem with general constant Dirichlet boundary conditions.

Affiliated with

Also associated with

No associations

Rating

Stable solutions for the bilaplacian with exponential nonlinearity does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Stable solutions for the bilaplacian with exponential nonlinearity, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Stable solutions for the bilaplacian with exponential nonlinearity will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-159698