Sub- and supersolution methods for prescribed mean curvature equations with Dirichlet boundary conditions

Abstract

We develop a sub-and supersolution method for prescribed mean curvature equations with Dirichlet boundary conditions. Our result is based on the work of Noussair, Swanson and Yang (1993) and essentially improve the method proposed by them. We remove some unnecessary assumptions from their main results, thus make the method much easier to use and broaden its range of applications. Further, we apply the improved sub-and supersolution method to several concrete examples such as a MEMS model, a Liouville-Gelfand type problem, and a sublinear problem. These applications give some new existence results and reflect the advantages of the improved method.

S-shaped connected component of radial positive solutions for a prescribed mean curvature problem in an annular domain

“…where Ω is a bounded domain in R N and the nonlinearity f : Ω × R → R is continuous, see [1,2]. The existence, multiplicity and qualitative properties of solutions of (1) have been extensively studied by many authors in recent years, see Coelho et al [3], Treibergs [4], Cano-Casanova et al [5], Pan et al [6], López [7], Corsato et al [8,9], Korman [10] as well as Ma et al [11,12], and the references therein. It is worth pointing out that the starting point of this type of problems is the seminal paper [13], and from Bartnik and Simon [2] as well as Bereanu and Mawhin [14], we know (1) has a solution whatever f is.…”

Periodic Solutions for a Prescribed Mean Curvature Equation with Multiple Delays

“…where ( ) = , ( ) = (1 + ) , and ( ) = − 1, respectively. For other recent developments and applications on the study of mean curvature equation, we refer the reader to [15][16][17][18][19][20], while the problem of periodic solution for prescribed mean curvature equation has been rarely studied [21][22][23][24]. Considering the delay phenomenon which exists generally in nature, Feng [22] studied the existence of periodic solutions for the one-dimensional mean curvature type equation in the following form:…”

Spectrum structure for eigenvalue problems involving mean curvature operators in Euclidean and Minkowski spaces

“…Obersnel and P. Omari [15], R. López [16], J. Mawhin [17], A. E. Treibergs [18], A. Azzollini [19], H. Pan and R. Xing [20] and references therein.…”