The Nontrivial Solutions For Some Fourth-order Semilinear Elliptic Problems

The solutions of many non-linear differential equations in the mathematical physics can be attributed to the critical point u of a functional I which is usually referred to Euler-Lanrange functional, in an appropriate Banach space, i.e. I’(u)=0. where I’(u) is the Frechlet derivative of the C1-functional I at the point u. Then, finding the critical points of functionals has become the key to solve the problem.This dissertation will primarily consider the existence of solutions for the fourth-order semilinear elliptic problem, by using Mountain Pass theorem and index theory under weaker conditions.The dissertation is divided into three sections according to contents.Chapter1Preference, we introduce the main contents of this paper.Chapter2In this chapter, we consider the following problem Where△2is the biharmonic operator, and Ω is a bounded domain in RN with smooth boundary (?)Ω, and c<λ1(λ1is the first eigenvalue of-△in H01(Ω)).In the present chapter, we will prove that there exist a positive solution and a negative solution and infinitely many solutions of problem (2.1.1), by using Mountain Pass theorem and Fountain theorem under weaker conditions. As a particular Linking theorem, Fountain theorem is a version of the symmetric Mountain Pass theorem. Of course, our results are still valid for second-order semilinear elliptic problem under weaker conditions.Chapter3In this chapter, let us consider the following problem Where△2is the biharmonic operator, and Ω is a bounded domain in RN with smooth boundary (?)Ω. In the last chapter, we have proved that there exist a positive solution and a negative solution and infinitely many solutions of problem (3.1.1), by using Mountain Pass theorem and Fountain theorem. But as c goes beyond λ1, the previous methods are no longer applicable because that the functional does not meet the mountain pass structure. In this chapter, we will study the solutions of problem (3.1.1) with c≥λ1, by using Linking theorem and index theory under weaker conditions.