Compactness along the branch of semistable and unstable solutions for an elliptic problem with a singular nonlinearity

Abstract

We study the branch of semistable and unstable solutions (i.e., those whose Morse index is at most 1) of the Dirichlet boundary value problem $-\Delta u = λf (x)/(1 − u)^2$ on a bounded domain $\Omega ⊂ R^N$ , which models—among other things—a simple electrostatic microelectromechanical system (MEMS) device. We extend the results of [11] relating to the minimal branch, by obtaining compactness along unstable branches for 1 ≤ N ≤ 7 on any domain \Omega and for a large class of “permittivity profiles” f . We also show the remarkable fact that powerlike profiles f (x) ≃ |x|^α can push back the critical dimension N = 7 of this problem by establishing compactness for the semistable branch on the unit ball, also for N≥8 and as long as $α>α_N =3N−14−4√6/4+2√6$. As a byproduct, we are able to follow the second branch of the bifurcation diagram and prove the existence of a second solution for λ in a natural range. In all these results, the conditions on the space dimension and on the power of the profile are essentially sharp

Touchdown and related problems in electrostatic MEMS device equation

“…Next, some estimates on λ are given for nonradially symmetric case [8,9]. Finally, existence of the non-minimal non-radially symmetric stationary solution is proven by the variational method [4]. Given u ∈ C λ , the linearized eigenvalue problem is defined as follows:…”

On the partial differential equations of electrostatic MEMS devices II: Dynamic case

“…Fine properties of steady states -such as regularity, stability, uniqueness, multiplicity, energy estimates, and comparison results-were also shown in [9] and [6] to depend on the dimension of the ambient space and on the permittivity profile. In particular, the following properties of positive minimal solutions of (S) λ were established.…”

Global existence and quenching for a damped hyperbolic MEMS equation with the fringing field

“…For an elliptic and parabolic operator A, thanks to the maximum principle, we have the results of the time-global existence [12,19,23,25] for sufficiently small λ > 0, the quenching [12,18,23,25] for sufficiently large λ > 0, the connecting orbit [23], the Morse-Smale property [23], the location of the quenching point [17] and its stationary solution [5,6,7,11,13,23]. Also in the hyperbolic problem, we have similar results to those in the parabolic case, i.e., the global existence [3,26,38], the quenching [3,26,32,38], the estimate of the quenching time [32] and the singularity of the derivative [2].…”