Results of Asymptotic Analysis of an Elliptic Equation
摘要
Suppose \(\Omega \subset \mathbb {R}^{2}\) a region, and consider the general elliptic equation \( \Delta w=g\left( w\right) , \) where g is a positive continuous function satisfying \( \lim _{w\rightarrow 0^{+}}g\left( w\right) =\infty . \) In the context of thin film equations, a solution w is classified as a point rupture solution, if there exists a point \(p\in \Omega \) , such that \(w\left( p\right) =0\) and \(w\left( p\right) >0\) in \(\Omega \backslash \left\{ p\right\} \) . We aim to analyze the asymptotic behavior of radial solutions. Specifically, our primary objective is to investigate how the limiting profile of radial solutions w and their corresponding energies vary as a function of the prescribed volume.