Some Properties Of Solutions For Quasi Linear Elliptic Equation

In the research of partial differential equation, it is very important to study quasilinear elliptic equations. Many problems in the physics can be attributed to quasilinear elliptic equations and systems. Whereas A-harmonic type equations are applied widely in quasiregular mappings and elasticity as well as physics.The well-poseness of solutions of qusilinear elliptic equations include existenc, uniqueness and stability. At first, people consider the classical solutions of equations, and then discuss the weak solutions and very weak solutions. There are some results about existence and uniqueness of weak solutions and very weak solutions for quasilinear elliptic equations of second order under under different spaces and different controlable conditions. And the stability of solutions is also the continuous dependence of solutions, which may be depended on the index of integrability, the boundary conditions, and other parameters.In the second chapter, the uniqueness of very weak solutions for equation-divA(x,Du)= f(x,u) with very weak boundary value is discussed on the basis of the known results. We obtain a uniqueness of very weak solutions by applying Hodge decomposition, combining with inequality of capacity and Sobolev inbedding theorem.In the third chapter, we discuss the stability of very weak solutions of equation-divA(x, Du)=f(x,u)with respect to the region. Sush stability of very weak solutions with regard to the region is obtained by using Hardy’s inequality, Vitali controlable convergence theorem as well as Sobolev inbedding theorem, and Poincare’s inequality.Sobolev space with variable index is mainly applied to research the growth problems with variable index, which stem from elasticity,electrorheology and image processing. These mathematical modes from the practical problems are constructed by the partial differential equation with variable index. And the existence, regularity of solutions are considered in the space with variable index.The relationship of the solutions of obstacle problems to equation div(p(x)|△u|p(x)-2△u)= 0 with the upper(lower) solutions of corresponding equation is discussed in the forth chapter.Some propties of solutions to obstacle problems of p-Laplace equation with constant index are extended to the case of obstacle problems for p(x)-Laplace equation with variable index, relations of the solutions of equation and its upper(lower) solutions are obtained.