Exact number of solutions of a prescribed mean curvature equation

Abstract

Keywords:Exact number of solutions One-dimensional prescribed mean curvature equation Time-map analysisWe study the exact number of positive solutions of the Dirichlet problem for the onedimensional prescribed mean curvature equationwhere λ > 0 is a parameter and p, q satisfy either 1 < p < q < +∞ or 0 < p < q < 1.

Existence of Positive Solutions of One-Dimensional Prescribed Mean Curvature Equation

“…Proof of Theorem 1. Similar to the proof of Theorem 1.1 [3], it follows from Lemmas 7 and 8 that the results are easy to prove.…”

“…Remark 3. If ( ) = max { , } and = , then (2) reduces to ( ) = , which has been considered by Habets and Omari [1]; ( ) = + for 0 < < < 1 and 1 < < < +∞, which has been studied by Li and Liu [3], and the case 0 < < 1 < < +∞ has been considered by Zhang and Feng [7].…”

“…where > 0 is a parameter and satisfies max { , } ≤ ( ) ≤ + , 0 < ≤ < +∞. The existence of positive solutions of such type of prescribed mean curvature equations both in one and in higher dimension has been discussed in the last decades by several authors (see Habets and Omari [1,2], Li and Liu [3], Pan [4], Obersnel and Omari [5,6], and others [7][8][9][10][11][12]) in connection with various configurations of nonlinearities.…”

“…Inspired by [1,3], we naturally consider (1) with nonlinearity satisfying max { , } ≤ ( ) ≤ + , 0 < ≤ < +∞.…”

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

“…One of the inequalities in (1.4) and (1.5) is strict, except for at most a finite number of z or u. and ϕ(s) = s √ 1+s 2 . Quasilinear problem (1.9) absorbed much attention in recent years and some special nonlinearities f satisfying f (0) = 0 such as u p , e u − 1, u p + u q , were studied and many interesting results on existence and exact multiplicity were obtained (see [1,2,4,5,6,8,11,12,14,15,19]).…”