Existence Of Periodic And Subharmonic Solutions For Some Second-order Hamiltonian Systems And Ordinary P-Laplacian Systems

Consider the following second-order Hamiltonian systemswhere T>0 and F:R×R^N→R satisfies the following assumption:(A′) F(t,x) is periodic in t with period T for each x∈R^N and F(t,x) is continuous fromR^N+1 to R withIn this paper, the minimax method in critical point theory and local linking theorem are employed to research the existence of nontrivial periodic solutions for nonautomous second-order Hamiltonian systems, and some new solvability conditions are obtained.The main results are the following theoremsTheorem 1 Suppose F satisfies (A′) and the following assumptions:(F1) There exist r₁>0,α<2Ï€²/T² such that0≤F(t,x)≤α|x|²for all |x|≤r₁,in R^N, t∈[0,T].(F2) There are constantsβ>0 and L>0 such thatF(t,x)≥β|x|2 for all |x|>L in R^N, t∈[0,T].(F3) There is a constantμ>2 such that Then system (HS) has at least a nontrivial solution.Theorem 2 Suppose F satisfies (A′), (F2), (F3) and the following assumptions: (F1′) There existsη>0 such thatF(t,x)≤F(t,0)for all |x|≤ηin R^N,t∈[0,T].Then system (HS) has at least a nontrivial solution.Furthermore, consider the following ordinary p-Laplacian systemswhere p≥2, T>0 and F:[0, T]×R^N→R satisfies the previous assumption (A′).Using the property of the uniformly convex Banach space and Generalized mountain pass theorem, we prove that the compact condition also in position under the assumptions of superquadratic condition, and some new solvability conditions are obtained.The main results are the following theorems.Theorem 3 Suppose F satisfies (A′) and the following assumptions:(F4) F(t,x)≥0, (?)_x∈R^N,t∈[0,T].(F5) There exist r₁>0,α<1/pC_p^-p such thatF(t,x)≤α|x|^pfor all |x|≤r₁ in R^N, t∈[0,T], where C_p=sup{‖u‖_L^p(?)‖_L^p=1,(?)=0}.(F6) There are constantsβ>0 and r₂>0 such thatF(t,x)≥β|x|^pfor all x|>r₂ in R^N,t∈[0,T].(F7) there existsμ>p such thatThen system (OPS) has at least a nontrivial solution in W_T^1,p.Theorem 4 Under the assumptions of (A′), (F4), (F7) and(F8) lim|x|→0 F(t,x)/|x|^p=0 uniformly for a .e.t∈[0,T](F9) There are constantsβ>3p^-1ω^p> and r₃ > 0 such thatF(t,x)≥β|x|^p for all |x|>r₃ in R^N, t∈[0,T], whereω=2Ï€/T.there exists a sequence (k_j)(?)N, k_j→∞, and corresponding distinct k_jT periodic solutionsof system (OPS).