A Study On A Class Of Nonlinear Elliptic Problems With Gradient Term

In this PhD thesis, we mainly study a class of quasilinear elliptic problems with gradient terms. As the nonlinearity f depends on the gradient of the unknow, this kind of quasilinear equations is not variational and the well developed critical point theory can not be applied directly. Many researchers fucus on how to obtain the existence and regularity of solutions of these problems. In this paper, we will try to get some results on these quasilinear elliptic equations.This PhD thesis is divided into six chapters. The first chapter is the introduction. In the first chapter, we will introduce what is elliptic partial differential equation, some results appeared about this class of quasilinear elliptic equations with gradient, innova-tion in this work and some research methods.The second chapter include some preliminaries. We will give some basic theories, lemmas and so on that we will need.In the third chapter, we mainly discuss the following elliptic boundary value prob-lemWe suppose that ? is symmetric with respect to x1, x2,……,xN-1 and f has the same symmetry with respect to x1, x2, ……,xN-1, then u(x) is (N-1)-radial sym-metric, that is u(x)=u(r,xN), where Thus we can transform the above equation into a elliptic equation about two variables, and by virtue of some methods dealing with elliptic equation about two variables, we are able to get the existence of solutions. Moreover, The solutions belong to C2,? (Q), but the solutions obtained in [70] is only of C1,?.In the fourth chapter, we mainly discuss the following elliptic boundary value prob- lem of nonlinear differential operator in divergence formWe will impose some conditions on A and the nonlinearity f, and prove the exis-tence of solution by virtue of the properties of monotone and coercive operator.In the fifth chapter, we will consider the following quasilinear elliptic problem with nonlinear Neumann boundary conditionsWe will try to apply a new iteration method the above equation. For each ??W1,p(?), we prove the existence of the mountain-pass solutions for the follow-ing equation which is variational then prove that the solutions convergence to a u, which is a solution of the original equation.At last, we give a conclusion and prospect about this paper.