Two Types Of Elliptic Equations With Biharmonic Operators

Elliptic partial differential equation is a kind of important partial differential equation.In recent decades,the study of elliptic equations has yielded fruitful results.It is widely used in fluid mechanics,elastic mechanics,electromagnetism and quantum mechanics.At present,the elliptic equation with double harmonic operator is still an important field in international mathematics,which has certain research significance.This paper mainly has two parts.The first part is devoted to a class of important and general nonlocal fourth order elliptic problemwhere ?~2= ?(?)is the bi-harmonic operator,? ? 0 is a constant.We focus on the case that f(x,u)involves a combination of convex and concave terms and the potential V(x)is allowed to be sign-changing.By new techniques,multiplicity results of two different type of solutions are established.Our results improves and generalizes that obtained in the literature.In the second part,we use the variational method to investigate nontrivial solutions(especially sign-changing solutions)for a class of fourth order elliptic problem with bi-nonlocal terms of the formwhich has received considerable attention in the past,?~2= ?(?)is the bi-harmonic operator and 1 <q < 2 <p <(2_*-2).As we all know that there is no maximum principle for the bi-harmonic problem.We will show our problem has least two nontrivial solutions of local minimum type and three nontrivial solutions of mountain pass type(one sign-changing)and infinitely many hign-energy sign-changing solutions.