The Study Of A Class Of Elliptic Equation With Negative Exponent Nonlinearity On The Boundary

The main purpose of this paper is to study a class of elliptic equations with negative exponent nonlinearity on a part of the boundary, which are derived from engineering and technical problems, such as MEMS. We mainly discuss the relationship between the exis-tence of solutions to the equation and the parameter A of the system, particularly we will estimate the maximal value A* of A, which is called pull-in voltage, ensuring the existence of solutions to the equation when the parameterλsatisfiesλ<λ*. In addition, we will discuss the properties of the solution when the parameter A satisfiesλ<λ* and the existence and uniqueness of solution to the equation in a certain weak sense when the parameter A satisfiesλ=λ*.The structure of the paper is as follows:In the first chapter, we will simply introduce the background of the MEMS, literature summary and major results; In the second chapter, we will prove the existence and the super bound of the pull-in voltageλ*; In the third chapter, we will discuss the properties of the minimal solution of the equation when the parameter A satisfiesλ<λ*; In the last chapter, which is also the most important part, we will study the properties of the extremal solution when the parameter A satisfiesλ=λ*.