Some Results And Methods In Studying Some Nonlinear Elliptic Equations Related To Fluid Theory

The development of nonlinear elliptic equations plays an important role in the study of partial differential equations. Semilinear and quasilinear elliptic equations are important branches of nonlinear elliptic equations.They have important applications in some fields of physics and chemistry,especially in the research of fluid theory in which some problems of the non-Newtonian fluids,non-Newtonian filtration and the bounded layer phenomena for viscous fluids can be resolved.Meanwhile,they also arise in the theory of heat conduction in electrical materials and chemical heterogeneous catalysts.Widely reading papers, learned the properties of different kinds of solutions,such as entire solutions,radial solutions,bounded solutions and explosive solutions.Besides,I have an understanding of the related theory of these solutions and the existence results.Some results and methods in studying semilinear and second-order quasilinear elliptic equations in recent years are described in this paper so that people can have some knowledge about them. The paper is organized as follows:(Ⅰ) In the first part, we introduce a class of semilinear elliptic equations of the type-Δu=λf(x, u),λ>0,among which the nonlinear term f(x, u) is allowed to be singular on the boundary. Whether the nonlinear term f(x, u) is in the pattern of p(x)g(u) divides this part into two sections.First,the solutions of a class of semilinear equations of the type-Δu= f(x, u)=p(x)g(u), in bounded domain,unbounded domain or N-dimension real space are discussed.The prob-lem arises in the study of non-Newtonian fluids. Many authors studied the existence of positive solutions and entire solutions with some methods such as the method of super-sub solutions,the comparison principle,the maximum principle,the fixed point theorem and so on.In addition,the equation-Δu=p(x)u-γ,x∈Ω(?)RN,where u satisfies homogeneous Dirichlet boundary conditions,has been extensively studied. Next,we introduce a class of semilinear elliptic equations of the type-Δu=λf(x, u),λ>0. The problem appears in the theories of fluid and heat conduction. Under some assump-tions,ground state solutions and positive solutions of the equations,where u satisfies Dirichlet boundary conditions,have been studied with the methods of a monotone convergence argu-ment and a fixed point theorem of generalα-concave operators.(Ⅱ)In the second part,we introduce a class of quasilinear equations of the type div(|▽u|p-2▽u)+f(x, u)=0.In the same way,whether the nonlinear term f(x, u) is in the pattern of p(x)f(u) divides this part into two sections.First,the solutions of a class of quasilinear elliptic equations of the type div(|▽u|p-2▽u)＋ρ(x)f(u)＝0, are discussed. The problem has been widely used in the study of non-Newtonian fluids,the turbulent flow of a gas in a porous medium,the subsonic motion of a gas and Riemannian geometry.The main researching method is the one of super-sub solutions. We introduce some results of singular solutions,ground state solutions,explosive solutions and radial solutions under some conditions.Next,we introduce a class of quasilinear elliptic equations of the type div(|▽u|p-2▽u)+f(x, u)=0. The problem has been extensively used in the study of non-Newtonian fluids and non-Newtonian filtration.We introduce some results of entire solutions,bounded solutions and positive solutions under some conditions,which are studied with some methods,such as the comparison principle,the maximum principle and so on.