Singularity of the extremal solution for supercritical biharmonic equations with power-type nonlinearity

Details Singularity of the extremal solution for supercritical biharmonic equations with power-type nonlinearity Singularity of the extremal solution for supercritical biharmonic equations with power-type nonlinearity

2011-01-20

arxiv.org/abs/1101.3903v2

Mathematics

Analysis of PDEs

13pages

Scientific paper

Let $\lambda^{*}>0$ denote the largest possible value of $\lambda$ such that $$ \{{array}{lllllll} \Delta^{2}u=\lambda(1+u)^{p} & {in}\ \ \B, %0\frac{n+4}{n-4}$ and $n$ is the exterior unit normal vector. We show that for $\lambda=\lambda^{*}$ this problem possesses a unique weak solution $u^{*}$, called the extremal solution. We prove that $u^{*}$ is singular when $n\geq 13$ for $p$ large enough, in which case $u^{*}(x)\leq r^{-\frac{4}{p-1}}-1$ on the unit ball.

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