Polyharmonic Equations With Exponential Nonlinearity

Recently, high order nonlinear partial differential equations have been paid more and more attention. This is because such equations have been widely applied to describe mod-els for thin elastic plates in classical mechanics, stationary surface diffusion flow, Hilfrich model in biophysics, and Willmore surface and Paneitz-Branson equation in differential geometry. On the other hand, during the mathematical study of such equations, various challenging problems have been put forward, and some new mathematical phenomena have emerged. In addition, this is a great opportunity for us to combine basic theory of partial differential equations, variational calculus, nonlinear analysis, geometry and mathematical physics, in order to solve practical scientific problems.In this dissertation, we will focus on the poly-harmonic partial differential equations with exponential nonlinearity. The first field of interest is radial symmetry of entire so-lutions to biharmonic equations. We investigat Δ2u ＝8S(N - 2) (N - 4)eu in RN with N≥ 5 and obtain sufficient conditions for an entire solution u to be radially symmetric. This result enriches the structure and properties of the solution set of higher order partial differential equations, and it is useful to study the geometric form of solutions.The second topic is the study of existence of solutions for high order conformally invariant equation. We investigat the polyharmonic problem △mu=±eu in R2m, with m≥ 2. In particular, we prove that for any V> 0, there exist radial solutions of △mu=-eu such that It implies that for m odd, given any Qo> 0 and arbitrary volume V> 0, there exist conformal metrics gu on R2m with constant Q curvature equal to Qo and vol9u (R2m)= V.The third topic is the study of classification of solutions to some elliptic equations. First we consider the polyharmonic equation (-Δ)mw= eu in RN with m≥ 3 and N> 2m. We prove that there exist many entire radial and non radial stable solutions, which implies that many new interesting phenomena appear, compared to the case of m= 2. Next we study the weighted nonlinear elliptic equation -div(|x|α▽u)=|x|reu, where a,r ∈ R satisfy N + a> 2,γ- α> -2, Liouville type results for the equation in RN are given.