Multiplicity Of Positive Solutions For Neumann Problem Of A Quasilinear Degenerated Elliptic Equation With Critical Growth

Now, for the broadly used in the practice, the theorem of partial differential equation be developed rapidly. And the character of the solutions of the elliptical partial differential equation is more kind, that make people to study it more broadly.In this paper we aims to study the Multiplicity of Positive Solutions for a Quasilinear Degenerated Elliptic Neumann Problem Involving Critical Sobolev Exponents as followingwhere Ω (?) R~N is the smooth boundary domain , u ≥ 0, 2 ≤ 2α < N, 0 < m <1, 2α < q~* - 1, q~* = (2αN)/(N-2a) .The thesis is composed of two chapters. The first chapter devote to the summarize of the dissertation, we recount about the development, background and the present circumstance of the second order elliptic partial differential equation.Also, we recount about some results we have obtained in the past two years in this chapter.The second chapter is the primary proportion of this paper which composed of four sections. The 2.1 section is in order to direct discuss the problem in the followed section, we give the conditions we need.In the 2.2 section, we discuss the necessarily lemmas and their proofs.In the next section, we mainly investigate the week continuous property of with the help of concentration-compactness principle.In the last section, with the help of the lemmas and conclusions we have obtained in 2.2 and 2.3 section, and the help of mountain pass lemma, we discussthe existence of positive solution of the initial problem. And by the use of Eke-land Variational Principle, we can obtain another positive solution, that we have complete the proof.