Hamilton Study Elliptic Equations Of Second Order Systems With A Class Of Fourth Order

We investigate the existence of periodic solutions and homoclinic orbits of the second order Hamiltonian systems, and the existence and multiplicity of solutions for a class of fourth-order elliptic equations in RN.The thesis consists of four chapters. Chapter1is the introduction.In Chapter2, we study the second order Hamiltonian systems where T>0, A(t) is a continuous symmetric matrix of order N. We consider the cases where the potential H is superquadratic at infinity and the potential H is subquadratic at zero. By using the fountain theorem of Bartsch and the symmetric mountain pass lemma due to Kajikiya, we obtain the existence of infinitely many periodic solutions, which unifies and improves some known results in the literature.In Chapter3, we consider the second order perturbed Hamiltonian systems u(t)-λL(t)u(t)+VW(t,u(t))=f(t), where λ≥1is a parameter, L∈C(R, RN2) is positive definite for all t R but unnecessarily uniformly positive definite for∈R, and W is either asymptotically quadratic or superquadratic in x as|x|â†'∞. Based on Ekeland’s variational principle and mountain pass theorem, we prove the existence of at least two nontrivial homoclinic solutions for the above system when f∈L2(R, RN\{0}) small enough.In the last chapter, we study the fourth-order elliptic equation where Δ2:=Δ(Δ) is the biharmonic operator, λ≥1is a parameter, V∈C(RN,R) and f∈C(RN×R,R). First, we consider the case where λ=1and V(x) satisfies:infx∈RN V(x)≥a>0. and for each b>0. meas{x∈RN:V(x)<b)<+∞, where meas denotes the Lebesgue measure in RN. Applying the symmetric mountain pass theorem, we prove the existence of infinitely many large-energy and small-energy solutions, which unifies and improves the recent result of Yin and Wu (J. Math. Anal. Appl.375(2011)699-705). Then, we consider the general case:V(x)>0for all∈MN and there exists b>0such that the set {x∈RN:V(x)≤b} has finite measure. We establish the existence and multi-plicity results when λ>1large.