M-Laplacian On Metric Space

With nonlinear problems being put forward frequently, we should study corresponding problems in larger spaces. The L^p spaces have many disadvantages in the field of application, and they have been generalized to Sobolev spaces W ^{1, p} and Orlicz-Sobolev spaces W¹ L_M.It's of importance to the study of partial differential equations. Many mathematicians subsequently investigate nonlinear equations and differential operators. In terms of the special p-Laplacian with good properties, many perfect results have been obtained. Metric spaces are worthy of research in mathematics. With the development of mathematicians'study,L^p spaces,Sobolev spaces,Orlicz spaces and Orlicz-Sobolev spaces have been generalized to metric spaces from Euclid spaces. Too much attention has been given to general nonlinear problems in newer and larger spaces. We introduce Orlicz-Sobolev spaces on metric spaces, and study their properties. We have further studied general m ? Laplacian based on them.In order to study the eigenvalue of the m ? Laplacian, it is necessary to study the properties of corresponding spaces, so we firstly discuss the theory of these spaces.In this paper, we prove the existence of the eigenvalues of in the Orlicz-Sobolev spaces on metric spaces. The existence of the eigenvalues will be obtained by the application of a general result about Lagrange multiplier which is due to Kubrusly. We shall minimize an appropriate functional subject to a constraint. We will find a function to satisfy in the Orlicz-Sobolev space with zero boundary values on metric space W₀¹ L_M(E) . X is a metric space, E(?)X...