Existence And Multiplicity Of Solutions For Some Hamiltonian Systems And Elliptic Boundary Value Problems

In this paper, by using variational methods, we study the existence of periodic solutions for Hamiltonian system, and existence and multiplicity of solutions for elliptic boundary value problem.Firstly, we consider the Hamiltonian systems-Ju-B(t)u=â–½H(t,u) (BHS) where B(t) is a symmetric 2N × 2N-matrix, continuous and T-periodic in t,H∈ C1(R × R2N,R) is a T-periodic function in t, is the standard 2N x 2N symplectic matrix, In is the N x N identity matrix, Some existence theorems are obtained for periodic and subharmonic solutions of a class of non-autonomous Hamiltonian systems. Our technical approach is based on a version of the Local Linking Theorem, and the Generalized Mountain Pass Theorem.Secondly, we study the p-Laplacian systems where p>1, T>0 and G:[0, T] x RNâ†'R is T-periodic in t for all x ∈RN. Two existence theorems are obtained for infinitely many periodic solutions of p-Laplacian systems (PLS) by minimax methods in critical point theory.Thirdly, we investigate the second-order systems where 1<p,q<∞,T>0 and |·| denotes the Euclidean norm in RN. F: [0,T]×RN×RNâ†'R. By applying the least action principle and the Saddle Point Theorem, two existence theorems are established.Later, we study the elliptic systems where Q is a bounded open subset of RN, with smooth boundary (?)Ω. and Hu denotes the partial derivative of H with respect to u. Using a version of the gen-eralized mountain pass theorem, we obtain the existence of nontrivial solutions for superquadratic elliptic systems (HES).In Chapter 6, we consider the Dirichlet boundary value problem where Ω(?)RN,(N≥3) is a bounded smooth domain, a ∈Lp(Q), p> N/2 and f∈C(Ω×R,R). By applying a version of the Local Linking Theorem and the Foun-tain Theorem, we obtain some existence and multiplicity results for superquadratic elliptic equations (EP).Lastly, we investigate the existence of solutions for the following p-Laplacian type equations with nonlinear boundary conditions where Ω(?) C RN is a bounded domain with smooth boundary (?)Ω,λ,μ∈R,1< p<N, f and g:Râ†'R are continuous functions. Here, the nonlinearity a Ω×RNâ†'RN fulfills certain structural conditions. The simplest case occurs when a(x,t)=|t|p-2t,p≥2, then (PNE) reduces to a p-Laplacian equation with nonlinear boundary conditions.a/av is the outer normal derivative. b(x) ∈ L∞(Ω) satisfies The existence of at least three solutions is established for p-Laplacian type equations (PNE). Our technical approach is based on a three-critical-point theorem.