Asymptotic Behavior of Rupture Solutions for the Elliptic MEMS Equation with H\'enon-Type and External Pressure Terms

TL;DR

This paper analyzes the asymptotic behavior of rupture solutions near the origin for a class of elliptic MEMS equations with Henon and external pressure terms, providing existence and detailed asymptotic expansions.

Contribution

It establishes the existence of radial and non-radial rupture solutions and characterizes their detailed asymptotic behavior near the rupture point.

Findings

Existence of radial and non-radial rupture solutions.

Full asymptotic expansion of solutions near the origin.

Characterization of asymptotic behavior for solutions.

Abstract

This paper investigates an elliptic MEMS-Type equation with Henon and external pressure terms: Delta u = lambda|x|^alpha / u^p + F for x in R^N \ {0}, with u(0)=0 and u>0 for x in R^N \ {0}, where N >= 1, lambda > 0, p > 0, alpha > -2 and F in R are constants. We study positive rupture solutions with rupture point at the origin (u(0)=0). Our main emphasis is on asymptotic radial rupture solutions: we prove the existence of both radial and non-radial solutions, characterize their asymptotic behavior near the origin, and obtain a full asymptotic expansion of arbitrary order.