Explosive solutions of elliptic equation with singular weight

Abstract

Some instances of explosive solution to elliptic equation of the form $\mathrm{\Delta }u=b\left(x\right)u^p-\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(x,\mathrm{\partial }\mathrm{\Omega }{)}^{-q}$ in a bounded smooth domain $\mathrm{\Omega }\subset R^N$ have been studied but only the case b is constant or $b\in C\left(\overline{\mathrm{\Omega }}\right)$ seem to have been considered up to now, at the best of our knowledge. In this paper, b is a positive weight function which can be singular on $\mathrm{\partial }\mathrm{\Omega }$. We obtain existence, uniqueness and exact boundary behaviour of positive solution.

Keywords:

• Elliptic equation
• explosive solution
• uniqueness

2010 Mathematics Subject Classifications:

• 35J25
• 35J75

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant number 11501199].