On the existence of positive solutions for some nonlinear boundary value problemsand applications to MEMS models

Hongjing Pan; Ruixiang Xing

doi:10.3934/dcds.2015.35.3627

Article Contents

2015, Volume 35, Issue 8: 3627-3682. Doi: 10.3934/dcds.2015.35.3627

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On the existence of positive solutions for some nonlinear boundary value problems and applications to MEMS models

Hongjing Pan^1, and
Ruixiang Xing^2,

1.
School of Mathematical Sciences, South China Normal University, Guangzhou 510631
2.
School of Mathematics and Computational Science, Sun Yat-sen University, Guangzhou 510275

Received: December 31, 2013

Revised: November 30, 2014

Published: May 2017

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Abstract

Motivated by some nonlinear models recently arising in Micro-Electro-Mechanical System (MEMS) and new progress on one-dimensional mean curvature type problems, we investigate the existence and exact numbers of positive solutions for a class of boundary value problems with $\varphi$-Laplacian $$ -(\varphi(u'))'=\lambda f(u)\; on (-L, L),\quad u(-L)=u(L)=0, $$ when the parameters $\lambda$ and $L$ vary. Various exact multiplicity results as well as global bifurcation diagrams are obtained. These results include the applications to one-dimensional MEMS equations with fringing field as well as mean curvature type problems. We also extend and improve one of the main results of Korman and Li [Proc. Roy. Soc. Edinburgh Sect. A, 140(6):1197--1215, 2010] (Theorem 3.4). With the aid of numerical simulations, we find many interesting new examples, which reveal the striking complexity of bifurcation patterns for the problem.

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Mathematics Subject Classification: Primary: 34B18, 34C23; Secondary: 35J93, 74G35.

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