The Nonlinear Boundary Value Problem Of A Class Of Elliptic Equations Involving The Critical Variable Exponent

In the studies of electrorheological fluid, nonlinear elasticity, rheology as wellas image restoration in practical applications, the classical Lebesgue and Sobolevspaces are inapplicable. Because such problems are inhomogeneous and nonlinearwith variable exponential growth conditions. So we need to study the problem basedon the theory of variable exponent Lebesgue and Sobolev spaces.In this paper, based on the theoryL^p（x）andW^1,p（x）spaces, we study thenonlinear boundary value problem of a class of elliptic equation involving criticalexponent.The main content of this article is divided into two parts.1. We consider the existence of the solution of the problem. We complete theproof by three steps under several conditions on the equation by using the“Mountain Pass Theorem”. First, we prove that the energy functional I exists （PS）sequence. Second we obtain that the （PS） sequence is bounded. Because of theequation involves the critical variable exponent, so theW^1,p（Ω） L⁽p*（x）^Ωis notcompact. So at last, we will use the principle of concentration compactness and thedefinition of the operator of typeS+and obtain the bounded （PS） sequence containsa convergent subsequence. Then we get the existence of the solution of the equation.2. We consider the existence of infnitely many solutions of this problem. First,we think over the critical point of the energy functional I by using “Symmetricalcritical point principle”.Second, we obtain the energy functional I contains asequence of critical value which tends to positive infinity by using “Fountaintheorem” The conclusion was established.