Second Order Hamiltonian Systems And Elliptic Resonant Boundary Value Problems

Periodic solutions of the second order Hamiltonian systems and the elliptic resonant boundary value problem are studied by the variational methods, the critical point theory and the implicit function theory. Some new solvability conditions and multiplicity results are obtained. The solvability conditions contain the subadditive condition; the subconvex condition; the locally coercive condition; a new Landesman-Lazer type condition and the sublinear condition. The main results include the following.First, consider the second order Hamiltonian systemwhere, T > 0, F : [0, T] × R~N → R satisfies the following assumption:(A) F(t,x) is measurable in t for every x ∈ R~N and continuously differentiable in x for a.e. t ∈ [0,T], and there exist a ∈ C(R~+,R~+),b ∈ L~1(0,T;R~+) such thatfor all x∈R~N and a.e. t ∈ [0, T].Suppose that α ∈ [0,1) and ▽F is α-sublinear, that is, there exist f,g∈ L~1(0,T;R~+) such thatfor all x ∈ R~N and a.ei ∈ [0,T]. Assume that Assumption (A) holds andThen problem (HS) has at least one solution.Suppose that F(t, x) satisfies Assumption (A) and is locally coercive, that is, there exist go G Ll(0, T) and a subset E of [0, T] with meas(￡) > 0 such thatF{t,x)>go(t)for all x G RN and a.ei G [0, T], and F(t.x) —> +oo as |x| —> oo for a.e.t G ￡*. Then problem (HS) has at least one solution. Suppose that F satisfies Assumption (A) andas |x| —> oo. Assume that there exists 7 > 1 such that F(t, x) is 7-subadditive in x for a.e. t G [0,T], that is,, x + y) < 7(F(t, x) + F(t, y)), Vi, y G ^.Then problem (HS) has at least one solution.Now we consider Dirichlet boundary value problem of the semilinear elliptic equationf —Au = XkU + g(x,u) VxGfi , ?,\u = 0 Vxedfl ^where ￡1 C RN(N > 1) is a bounded smooth domain, A^ is the k-th distinct eigenvalue of the eigenvalue problem-Au = Xu Vx G fi, u = 0g : Q x R —>? R is a Caratheodory function. Suppose thatfor all M > 0 and some q>\ such that ^ + - = 1, where 2 < p < -^ for N > 3, 2 < p < +00 for N = 1, 2. Assume that there exist r g]1, 2[,a GLl(Q) with a{x) > 0 for a.e. x E ￡1 and Jha(z)dx > 0, and b E Z/(Q) suchthat, . ^ ,. . . q(x,t)t , q(x,t)t , . .a(x) < lirninf ^J- < limSup^-^- < b(x)uniformly for a.e.z E ft, where -^ + r = 1. Then problem (D) has at least one solution in Hq (Q).Suppose that g E C(R, R) such that|t|->oc t \t\-*oo tAssume that h E Ll(d,7r) satisfiesF{—oo) smxdx < h(x)smxdx < F(+oc) / sinxcta where F(—oo) = HmsupF(t). F(+oo) = liminf F(￡) andt—>—oot—>+oo-J Vo9(s)ds-g(t) t^OThen problem-u" = u + g(u)-h(x). u(0)=u{tt)=0has at least one solution in Hq(0,tt).At last we consider Neumann boundary value problemdu- Au = f{x,u) + eh(x) infi, 77-=° ononwhere tt C RN(N > 1) is a bounded domain with a smooth boundary and outward normal n(x) and du/dn = n(x) ■ Vu, / : fl x R —> R, f(x,t) is measurable in x for all t E R and continuously differentiate in t for a.e. x E fl and there exist a constant C\ > 0 and 2 < p < 2N/(N - 2) for AT > 3(2 < p < +00, for N = 1,2) such thatfor all t G R and a.e. x G Q., where f't(x,t) = df/dt. Assume that there exist 5 > 0 and a,b G L°°(Q.) such that a(x) > /im, b(x) < /J,m+\ for a.e. x G fi,meas {x G n|a(a:) > /^m} > 0, meas {x G Q\b(x) < fj,m+i} > 0,anda{x) < ^^ < b(x)for all 0 < |t| < S and a.e. x G fi, where /im is the m-th distinct eigenvalue of the eigenvalue problemdu— A u = uu in Q, — =0on 5fi.cmand that /i G L2(Q,) satisfyingh = (meas SI)"1 / h(x)dx — 0.LetF(z,￡) -? -ooas \t\ —> oo uniformly for a.e. iGfl. Then for small ￡, problem (iV) has at least three distinct solutions in Hl(Q).