Existence Of Solutions Of A Class Of Quasilinear Elliptic Equations

In this paper, we discuss the existence and multiplicity of solutions for a class of quasilin-ear elliptic equations involving the operator div(a(x, (?)u)). Firstly, in the case of subcritical nonlinearities, by means of constructing the local minimum of functional corresponding to the equation, we prove the existence of multiple nonnegative solutions to the equations under the Neumann boundary condition. Moreover, the sequence of the solutions is unbounded or convergent to 0.Secondly, in the case of nonlinearities involving the critical Sobolev exponent and of the operator div(a(x, (?)u)) being the prescribed mean curvature operator for a nonparametric surface, by using the concentration-compactness principle due to P. L. Lions and some other variational methods, we prove the existence of nontrivial solutions of the equations under the Dirichlet boundary condition.Finaly, in the case of the nonlinearities being singular at 0, by utilizing the mountain-pass theorem and some other variational methods, we get the existence of nontrivial solutions of the equations under the Dirichlet boundary condition, where the operator div(a(x, (?)u)) is the p-Laplacian.