The Solvability For Several Classes Of Elliptic Equation Boundary Value Problem

This article is divided into four chapters, the first chapter studies existence of weak solution the classical solution for a class of semi-linear elliptic equations boundary value problems in a domain with a hole. Second and third chapters by introducing new variables to polyharmonic equation boundary value problem is transformed into elliptic equation system boundary value problems, and by using of the maximum principle, the continuum theory and a fixed point theorem,we studied the existence of solutions and the existence of positive solutions of polyharmonic equations boundary value problems. The fourth chapter studies the equivalence for variational problem and biharmonic equation boundary value problem.The first chapter studied the solvability of boundary value problems for semi-linear elliptic equations where Ω(?)RN is a bounded domain with a hole, the inner boundary Γ2, outer boundary Γ1, and δΩ=Γ1∪Γ2smooth, b>0is a constant.In this chapter, using upper and lower solutions method, embedding theorem and Leray-Schauder fixed point theory proved the existence of weak solutions for a class of semi-linear elliptic equation boundary value problems in a domain with a hole. We also obtained the existence of classical solution by using of the Schauder fixed point theory and upper and lower solutions method.The second chapter studied the biharmonic equation boundary value problems where Ω is a bounded smooth domain in RnIn this chapter, we use the maximum principle and the continuum theory proved positive solution existence for this kind of biharmonic equation boundary value problem,The third chapter studies the solvability of polyharmonic equation boundary value problems where Ω is a bounded smooth domain in RnIn this chapter, by introducing a new variable polyharmonic equation boundary value problem is transformed into elliptic equation system boundary value problem, we studied existence and nonexistence of solution by using of the fixed point theorem.The fourth chapter we proved the equivalence quasilinear biharmonic equation boundary value problem, quasilinear polyharmonic equation boundary value problem and variational problem by using of the variable method basic principe.