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August  2015, 35(8): 3627-3682. doi: 10.3934/dcds.2015.35.3627

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

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

Received  January 2014 Revised  December 2014 Published  February 2015

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.
Citation: Hongjing Pan, Ruixiang Xing. On the existence of positive solutions for some nonlinear boundary value problems and applications to MEMS models. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3627-3682. doi: 10.3934/dcds.2015.35.3627
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