Classification Of Positive Radial Solutions And Eigenvalue Problem For Semilinear Bi-Laplace Equation

Recently,more and more attention is paid to the study of fourth-order elliptic partial differential equations.On the one hand,the fourth-order elliptic equations have been widely used in geometry,physics,mechanical engineering,such as the Paneitz operator and Willmore surfaces in geometry,Hilfrich model in biophysics,clamped plates in classical mechanics,etc.On the other hand,the properties of fourth-order elliptic equations and second-order elliptic equations are very different.For example,in general fourth-order elliptic equations no longer satisfy the maximum principle.So mathematically speaking,this brings us some difficulties,but also brings us some new mathematical phenomena.In this dissertation,we mainly discuss some qualitative properties of several classes of fourth-order elliptic equations.In the first part,we consider the weighted fourth-order equationΔ(|x|-αΔu)+λdiv(|x}-α-2(?)u)+μ|x|-α-4u=|x|βup in Rn\{0},where n≥ 5,-n<α<n-4,p>1 and(p,α,β,n)belongs to the critical hyperbola We prove the existence of positive radial solutions to the above equation for someλ and μ,and we use the Emden-Fowler transformation v(t):=|x|n-4-α/2u(|x|),t=-ln |x| to classify these positive radial solutions into three types.On the other hand,for the radial solution u with non-removable singularity at origin,v(t)is a periodic function if α∈(-2,n-4)and λ,μ satisfy some conditions;while forα∈(-n,-2],there exists a radial solution with non-removable singularity and the corresponding function v(t)is not periodic.We also get some results about the best constant and symmetry breaking.In the second part,we consider the bounds of the Dirichlet eigenvalues for the equation where Ω is a smooth bounded domain in Rn with 0∈Ω,n≥ 4 and μ≥μ0:=-n2(n-4)2/16.Assume {λμ,i}i∈N are the eigenvalues of the above equation with 0<λμ,1≤λμ,2≤…≤λμ,k≤…,we provide lower bounds of the k-th eigenvalue and upper bounds of the(k+1)-th eigenvalue.In the last part,we prove a Liouville type theorem for the smooth stable solutions to the equationΔ2u=|u|p-1u+|x|a|u|q-1u in Rn,where n≥ 5,a≥0,q∈(1,qc(n))(the definition of qc(n)will be given later)and p,q satisfying 4/p-1=4+a/q-1.First,we prove a monotonicity formula about the solution u of the above equation,and then we prove the non-existence of homogeneous stable solution.Finally,we prove the non-existence of smooth stable solution by some integral estimates and the monotonicity formula.