Bifurcations of some elliptic problems with a singular nonlinearity via Morse index

Zhang, Guo; Liu, Zhongyuan; Wei, Juncheng; Zhou, Feng

doi:10.3934/cpaa.2011.10.507

Cited by 7 publications

(1 citation statement)

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“…Based on techniques developed in [1], the authors in [9] obtain the existence of infinitely many turning points for −∆U = Λ|x| α /(1 − U) p . However, the locations of turning points were not obtained and in particular, the infinite multiplicity of classical solutions at Λ = Λ * was not proved (see also [7]). In [13] a similar conclusion to that in Corollary 1.2 above is obtained except the uniqueness of the degenerate solution for the equation −∆ m U = Λ|x| α /(1 − U) p .…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Radial single point rupture solutions for a general MEMS model

Ghergu¹,

Miyamoto²

2020

Preprint

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We investigate the structure of radial solutions corresponding to the equation0) = 0 and f satisfies certain assumptions which include the standard case of pure power encountered in the study of Micro-Electromechanical Systems (MEMS). A particular attention is paid to degenerate solutions of the above equation, that is, solutions u * which are positive in B r0 \ {0} and vanish at the origin. We prove that a degenerate solution u * exists, is unique and equals the limit of the regular solutions u( • , α) (with u(0, α) = α) in C 2 loc (0, r 0 ) ∩ C loc [0, r 0 ) as α → 0. The rate at which u * approaches zero at the origin is also obtained. Further, we show that the number of intersection points between u * and u( • , α) tends to infinity as α → 0. This leads to the complete bifurcation diagram of MEMS type problems.

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