Existence And Multiplicity Results For Some Quasilinear Elliptic Systems

This dissertation is the main results obtained by the author during the period when she has applied for the M.D.The contents are as follows:In chapter two, we study the existence of solutions for the following quasilinear elliptic systems involving the (p(x), q(x))-Laplacian. The main difficulty of the problem is that the (p(x), q(x))-Laplacian operator possesses more complicated nonlinearities than the (p, q)-Laplacian. We study this equation systems on a bounded domain, and obtain three solutions under appropriate hypotheses on the potential functions. Our technical approach is based on the general three-critical-points theorem obtained by B. Ricceri.Chapter three studies the existence of the solutions for the Gradient elliptic sys-tems with nonlinear Neumann boundary conditions However, in this Chapter we will restrict ourselves to the case that F(x, u, v) satisfies the subcritical growth condition because the study of the (p(x), q(x))-Laplacian equa-tions in the critical growth case is very difficult and requires some special preliminaries which are not ready up to the present. Here, we also give suitable assumptions on the potential F(x, u, v), and by using the uniform convexity of the Sobolev space, we ob-tain the weak solution of the equation systems. Moreover, the nontrivial solution of the problem is considered.