Bulletin of the Korean Mathematical Society (대한수학회보)

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A NOTE ON BOUNDARY BLOW-UP PROBLEM OF 𝚫u = up

  • Kim, Seick (Department of Mathematics Yonsei University)
  • Received : 2018.03.13
  • Accepted : 2018.05.29
  • Published : 2019.01.31
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Abstract

Assume that ${\Omega}$ is a bounded domain in ${\mathbb{R}}^n$ with $n{\geq}2$. We study positive solutions to the problem, ${\Delta}u=u^p$ in ${\Omega}$, $u(x){\rightarrow}{\infty}$ as $x{\rightarrow}{\partial}{\Omega}$, where p > 1. Such solutions are called boundary blow-up solutions of ${\Delta}u=u^p$. We show that a boundary blow-up solution exists in any bounded domain if 1 < p < ${\frac{n}{n-2}}$. In particular, when n = 2, there exists a boundary blow-up solution to ${\Delta}u=u^p$ for all $p{\in}(1,{\infty})$. We also prove the uniqueness under the additional assumption that the domain satisfies the condition ${\partial}{\Omega}={\partial}{\bar{\Omega}}$.

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References

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