Semilinear Elliptic Problems And Solutions Of Second Order Hamiltonian Systems

The paper is divided into two chapters to discuss. In the chapter1we will consider the existence of a sign changing solution to the semilinear elliptic problem-â–³u+u=|u|p2u, u∈H10(Ω) in an exterior domain Ω having finite symmetries under Cerami condition.Section1,in this introduction, we will deal with the existence of solution for the following problem And obtain the variational functional Jo:H01(Ω)â†'R associating to the problem(1.1); It gives (g1)-(g3) and the A.R. condition of2-superlinear case,what’s more, we give an example to prove that (g1)-(g3)is weaker than the A.R. condition. At last,under (g1)-(g3),we will introduce Theorem1.1, Theorem1.2which to be proven.Section2,it mainly introduces an orthogonal action on H01(Ω),the fixed point space of this action and the radial projection onto the Nehari manifold and so on.Section3,it mainly introduces Cerami sequence,(C)Φc condition and corresponding Lemmas and Corollary.Section4,the proof of Theorem1.1.Section5,the proof of Theorem1.2.In the chapter2, for a class of the second order Hamiltonian systems by using the local linking lemma, we will obtain at least two nontrivial homoclinic orbits, where W(t,u) is superquadratic.The chapter2is divided into two sections according to contents. Section1Preference, in this section,we will introduce Consider the second order Hamiltonian systems Besides, under the case thatL∈C(R,RN2) and W∈C1(R×RN, R) satisfy (L1)-(L2)and (S1)-(S4) seprately,we will introduce the theorem that will be proved.Section2In this section, at first,we will introduce the definition of self-adjoint and the spectral resolution self-adjoint. Denote D=D(|A|1/2),and in D,we define inner product and norm;prove that the spectrum σ (A) consists of eigenvalues numbered and a corresponding system of eigenfunctions,denote E-:=span{e1,...,en-}, E0span{en-+1,..., en}:=ker A, E-:=span{en+1,...}, and E=E-(?)E0E(?)E+. In E, we introduce inner product and corresponding norm.Then,we will introduce the definition of the function f∈C1(X, RN) having a local linking at0, with respect to (X1,X2) and f satisfying (PS)*condition,as will asLemma2.3.([6]) If L satisfies (L1) and (L2). D(A) is continuously embedded in W2,2, andLemma2.4.([3]) Suppose that f∈(X, R) satisfies the following assumptions:(B1)f has a local linking at0and X1={0},(B2)f satisfies (PS)*(-B3)/map bounded sets into bounded sets,(-B4)/is bounded below and d:=infxf<0.Then f has at least three critical points.Finally,we will prove the above theorem in five steps. Step1proves that f has a local linking at0and X {0};Step2proves that f satisfies (PS)*;Step3proves thatf bounded below,and d infxf<0;Step4proves thatf map bounded sets into bounded sets;Step5proves that (?)u∈D(A) is the solution of (1.1).