Existence And Multiplicity Of Solutions For Some Second-order Hamiltonian Systems And Semilinear Elliptic Equations

Consider the non-autonomous second-order Hamiltonian systemswhere T > 0, ω = 2Ï€/T, m is a non-negative integer and F : [0,T]×R^N → R satisfies the following assumption:(A) F(t, x) is measurable in t for each x ∈ R^N and continuously differentiable in x for a.e. t ∈ [0,T], and there exist a∈ G C(R⁺,R⁺),b ∈ L¹(0,T;R⁺) such that|F{t,x)| + |F(t,x)|N and a.e. t∈ [0,T].When m=0, Consider following non-autonomous second-order Hamiltonian systemswhere T > 0, F :[0,T}×R^N → R satisfies conditions (A).In this paper, the existence and multiplicity results of solutions are obtained by the reduction method for non-autonomous second order Hamiltonian systems, using the least action principle, local linking in critical point theory. The main results are the following theorems.Theorem 1 Assume that R satisfies assumption (A), If the following conditions hold:(a) there exists a function such that - monotone, that is,for all x,y ∈ R^N and a.e. t ∈ [0,T];(b) there exists f,g G with such thatfor all and a.e. t [0,T].Then the system {HS₁) has at least one solution in H_T¹.Theorem 2 Assume that F:[0,T] satisfies assumption (A), conditions (a), (6) and the following condition:(c) there exists δ > 0 and an integer l > m such thatfor all |x| < 6 and a.e. * G [0,T];Then the system {HS\) has at least a nonzero solution in Hj,.Theorem 3 Assume that F:[0,T] x RN -> R satisfies assumption {A), conditions (i), (u) and the following condition:(c)' there exists 6 > 0 and an integer / > m such thatimV|x|2 - \(l + l)V|x|2 < F?,x) - F(t,0) < \mW\x\2 - |/V|i|2for all |x| < 5 and a.e. i G [0, T};Then the system (HSi) has at least two nonzero solutions in H\,.Theorem 4 Assume that F:[0,T] x RN -> R satisfies assumption (A), and that there exist 7 G Ll(0,T;R), and 0 < ï¿¡-y(t)dt < 12/T, such that(VF(*,x) - VF(t,y),x- y) > -7(0l* - y|2for all x,y G RN and a.e. t G [0,T].Suppose that fï¿¡ F(t,x)dt -> -oo, as |x| -> +oo. Then the system (/fS2) has at least one solution in H\,.Theorem 5 Assume that F:[0,T] x RN -> R satisfies assumption (A), and that there exist h.-y G L'(0,T;/l) and jj i{t)dt < ^, f^' h{t)dt > 0, such that-7(01* - y\2 < (VF(t,i) - VF((,y),i- y) < -/i(0|x - j/|2 for all x,y G /^'v and a.e. t G [0,T]. Then the system (HS?) has at kvust one solution in IIj..er, we consider the elliptic equations with Dirichlet boundary conditionI u = 0 on c^fi;where L* C R'W{N > 1) is a bounded domain, / : il x 7? -> /i is a Carathcodory function, that is, /(x,i) is measurable in i for every t & R and continuous in ï¿¡ for a.e. x G $2, and h(x) G //-'(fi).Let \k{k 1,2,...) be the A:-th distinct eigenvalue of the eigenvalue problem—Au — Xu iii il, u ■— 0 on dtt.and E(\k)(k = 1,2,...) be the eigenspace corresponding to A^.Theorem 6 Assume that there exist a constant C > 0 and a real function 7 G Ll (0) such that\f(x,t)\{+1(x). (1)for all t G R, a.e. x G Q, where 2' = 2N/(N - 2);if TV > 3 and 2* may be replaced by any number in [1, +00) if N = 1 or TV = 2, and that^f) (x) > 0 for all x G Q. Then the problem {EP) has at least one solutions in Hq(U).Theorem 7 Assume that there exist Co > 0 and 2 < p < 2N/{N - 2) for TV > 3(7;G (2, +00) for TV = 1,2) such that\f(x,t)\ 1 and (5 > 0 such thatAm|?|2<2F(x,0 A* , suchthat/(?,?)-/(?,0 (6)s — t for all 5, t G R, s ^ t and a.e. x G fi, and assume that2F(i,<) limsup—^— < a(x) < Xk+X (7)for a.e. x G ft, with the second inequality's strict inequality hold on a set of positive measure E C H. Suppose that /? = 0. Then the problem (EP) has at least one solutions in Hq(Q).Theorem 9 Assume that / satisfies (4). If the conditions (6)(7)hold, and there exists an integer m> k + 1, S > 0 such that(8)for all \t| < 6 and a.e. iGQ. Suppose that h = 0. Then the problem {EP) has at least two nonzero solutions in Hq(ï¿¡1).Theorem 10 Assume that / satisfies (4), h = 0 and there exists an a < A*+i such thatf{x,s)-f(x,t) for all x, t G R, s ^ t, and a.e. iGfl. Assume thatliminf 2ir^ >b(x)>Xkfor a.e. x G ft, with the second inequality's strict inequality hold on a set of positive measure Ecil.Assume that there exist 6 > 0, m < k such thatfor all |(| < 6 and a.e. i6fl. Then the problem {EP) has at least one solutions in...