Study on the behaviors of rupture solutions for a class of elliptic MEMS equations in \R²

2023/10/27 by Qing Li, Li, Qing, Yanyan Zhang +1 · 1 citation
Mathematics · Engineering · #Advanced Mathematical Physics Problems #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2310.18140

Abstract

This study examines nonnegative solutions to the problem \\arraycolsep=1.5pt \begin arraylll Δu=(λ|x|^α)/(u^p) amp;\hbox in \R ²∖ \0\,
u(0)=0 and ugt; 0 amp;\hbox in \R ²∖ \0\,
\endarray. where \lam >0, \alp>-2, and p>0 are constants. The possible asymptotic behaviors of u(x) at |x|=0 and |x|=∞ are classified according to (α,p). In particular, the results show that for some (α,p), u(x) exhibits only ``isotropic" behavior at |x|=0 and |x|=∞. However, in other cases, u(x) may exhibit the "anisotropic" behavior at |x|=0 or |x|=∞. Furthermore, the relation between the limit at |x|=0 and the limit at |x|=∞ for a global solution is investigated.

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