Positive Solutions For Some Critical Growth Nonlinear Elliptic Equations With Neumann Boundary Conditions

Nonlinear elliptic equations involving critical Sobolev exponent arise in Ri-mannian manifold and conformal geometry,as well as in the (?)id mechanics.They have broad application in the research of elastomer and conduit's fluid.The existence and regularity of the solutions also give the theoretical support in the numerical approach and calculations. Now it is popular to study such problems in the field of elliptie equations by the variational method and the theory of critical points.Pohozaev had the result of Pohozaev identity in 1965.Further-more,A.Ambrosetti and P.H.Rabinowitz gave the Mountain Pass lemma(1973) , and many new minmax principles.After that,people got a lot of significant re-sults.Concerned to the nonlinear elliptic equations which contained the Lapace operator or p-Lapace operator,In different dimension and domain,they mostly consider the existence and regularity of solutions with Dirichlet boundary condition or Neumann boundary condition.In this paper.we discuss the singular critical growth nonlinear elliptic equation,i.e.- Δu =(?) f(x, u)with the Neumann boundary condition.In the theoretical basic of weak convergence and critical points,we prove the existence of solution in the Sobolev space H~1(Ω) and we get the field of the critical value.This paper is divided into 5 parts.In the first part.we give a whole introduction to the problem and then present my own problem.In the second chapter, we introduce some knowledge which we'll use after.In the third chapter,we give some useful lemmas and their proofs.We especially consider the case of f(x,u) = —λu.In the use of Mountain Pass lemma,we get the theory of solutions'existing.In the fourth chapter,we discuss the general case.and prove the existence of solutions.After that,we got two useful corollaries. At last,based on the paper,we give some my own opinions about the future work.