Two Nonlinear Elliptic Equations Existence

In this paper we investigate multiple solutions for a semilinear problem and pos—itive solutions of some fourth order elliptic equation.In chapter1,multiple solutions for the problem(?) is discussed where Ω(?) RN is a smooth bounded domain,N≥p+1,1≤q<p and h belongs to an appropriated Lebesgue space. By discussing the eigenvalue problem under Neumann boundary,we obtain multiplicity of solutions when f is an asymptotically linear function and the norm of h is small.In chapter2,we study the following biharmonic problem:(?) where0<q<1<p=N+4/N-4,Ω(?)RN(N>4)is a bounded domain with smooth boundary (?)Ω,and f,g are nonnegative continuous functions on Ω.This chapter discusses the positive solutions of the equation when f(x)and g(x)satisfy suitable conditions.Basing on the extraction of Palais-Smale sequences in Nehari manifold, the difficulty of this chapter is the existence of a positive ground state solution of the equations.And the existence of at least two positive solutions is obtained for the equation for sufficiently small入.