Existence Of Positive Solutions Of Semilinear Biharmonic Equations

We consider the semilinear biharmonic problem Where Δ2=-Δ(-Δ) is a biharmonic operator, Ω, is a bounded domain with smooth boundary in RN,N≥5. We prove the conclusion as follows.In this paper, we suppose that/satisfies the following: (H1)f∈C2(Ω×R), and f(x,t)≥0 if t≥0; (H2)f(x,0)=ft’(x,0)=0 for all x∈Ω; (H3)there exist T>0 and 1<p<(N+4)/(N-4) such that |ft’(x,t)|≤C|t|p-1,|▽xf(x,t)|≤C|t|p,|▽xft’(x,t)|≤C|t|p-1; for |t|≥T and x∈Ω; (H4)there exist μ> 0 and T>0 such that f(x, t)≥μtp for all x∈Ω and t≥T.Then problem (*) has at least a positive solution.This paper is mainly composed of three chapters:Chapter 1 is the introduction. We introduce the research background of this article and the development of semilinear biharmonic equations, then we give the main theorem.In chapter 2, we prove some new nonlinear Liouville type theorems which may be usefull in other situations.In chapter 3, we prove Theorem 1.1 with the truncation technique, the mountain pass lemma and the blow-up methods.