Solvability Of Boundary Value Problems For Nonlinear Elliptic Equations(Systems)

In this paper,we investigate the solvability of nonlinear elliptic equations and systems by using the fixed point theorem,the sub-super solution method.The introduction mainly introduces the development history and background of partial differential equations,as well as the methods used in this paper.The first chapter studies the solvability of the boundary value problem of biharmonic equations with small parameter ?.(?)(1.1)Here the ?(?)Rn is a bounded smooth cavity region,where the inner boundary is?2,the outer boundary is ?1.And is a constant,? is a positive parameter.In this paper,we use variable substitution in the problem(1.1),set-?u=v,and convert the problem(1.1)into a boundary value problem of an elliptic system of equations(?)(1.2)Then,we use the sub-super solution method and the fixed point theorem to prove the existence of the solution to the above problems,and discuss the uniqueness of understanding.The second chapter discusses the solvability of boundary value problems for semilinear elliptic systems(?)(2.1)Here the c(x),d(x)is a continuous positive function in ?,c(x)>0,d(x)>0,?,? ?(1,?)are constants.In this paper,the existence of the solution to the problem(2.1)is studied by using the fixed point theorem.Finally,the uniqueness of the solution is proved by using Green identity and the extremum principle of harmonic functions.The third chapter,we investigate the boundary value problem of semi-linear elliptic equations on a bounded hole region#12Here k>1 is a constant,Q(?)Rn is a bounded smooth cavity region,where the inner boundary is ?2,the outer boundary is ?1,?1 and ?2 are positive parameters,b>0 is a constant,(?).In this paper,we prove the existence of the solution by using the method of sub-super solution method.Finally,we consider a special case,that is,when ?2 is a constant,we prove the existence of the solution.