Asymptotic Behavior of Rupture Solutions for the Elliptic MEMS Equation with Hénon-Type Term

TL;DR

This work analyzes rupture (touchdown) solutions of the elliptic MEMS equation with a Henon-type term near a rupture point at the origin. It develops a systematic asymptotic framework, first for radial ruptures and then for nonradial ruptures that are asymptotically radial, yielding full arbitrary-order expansions with explicit leading profiles and angular corrections. The authors establish existence of these rupture solutions using a fixed-point approach in weighted Hölder spaces, leveraging a transformed variable formulation and angular decompositions into spherical harmonics. Collectively, the results extend isotropic analyses to Henon-type inhomogeneities and provide a detailed description of both isotropic and anisotropic rupture behavior, with implications for touchdown dynamics in MEMS devices.

Abstract

For an elliptic MEMS equation with Hénon-type term, $Δu = λ|x|^α/u^{p} + F$, we study rupture solutions, i.e. solutions for which $u(x_0)=0$ at some point $x_0$, also $x_0$ is called a rupture point. In this paper we focus on the special case where the rupture occurs at the origin. According to the different Hénon-type exponents $α$, we analyze the asymptotic behavior of such solutions near the origin and derive a full asymptotic expansion of arbitrary order in a neighborhood of the origin. Moreover, for both radial solutions and non-radial solutions with asymptotic radial condition, we prove the existence of rupture solutions near the rupture point by constructing it.

Figures (1)

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Theorems & Definitions (18)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Theorem 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5
• Lemma 3.6
• Lemma 4.1