Kazunaga Tanaka

Some results on connecting orbits for a class of Hamiltonian systems.

Abstract : The existence of various kinds of connecting orbits is established for a certain Hamiltonian system as well as its time dependent analogue. For the autonomous case, our main assumption is that V has a global maximum, e.g. at X = O and we find a various kinds of orbits terminating at O. For the time dependent case V has a local but not global maximum at X = O and we find a homoclinic orbit emanating from and terminating at O. Keywords: Periodic solution.

A remark on least energy solutions in RN

We study a mountain pass characterization of least energy solutions of the following nonlinear scalar field equation in R N : -Δu=g(u), u ∈ H 1 (R N ), where N > 2. Without the assumption of the monotonicity of t → g(t)/t, we show that the mountain pass value gives the least energy level.

Trudinger type inequalities in ^{} and their best exponents

We study Trudinger type inequalities in RN and their best exponents αN . We show for α ∈ (0, αN ), αN = Nω N−1 (ωN−1 is the surface area of the unit sphere in RN ), there exists a constant Cα > 0 such that ∫ RN ΦN α( |u(x)| ‖∇u‖LN (RN ) ) N N−1  dx ≤ Cα ‖u‖NLN (RN ) ‖∇u‖N LN (RN ) (∗) for all u ∈W 1,N (RN ) \ {0}. Here ΦN (ξ) is defined by ΦN (ξ) = exp(ξ) − N−2 ∑ j=0 1 j! ξ . It is also shown that (∗) with α ≥ αN is false, which is different from the usual Trudinger’s inequalities in bounded domains. 0. Introduction In this note, we study the limit case of Sobolev’s inequalities; suppose N ≥ 2 and let D ⊂ R be an open set. We denote by W 1,N 0 (D) the usual Sobolev space, that is, the completion of C∞ 0 (D) with the norm ‖u‖W 1,p 0 (D) = ‖∇u‖p+‖u‖p. Here ‖u‖p = (∫

Morse indices at critical points related to the symmetric mountain pass theorem and applications

We obtain estimates of Morse indices of an even functional at critical points related to the Symmetric Mountain Pass Theorem ([AR, R1, R4]). As applications, we deal with the existence of multiple solutions of inhomogeneous super linear boundary value problems and improve the result of [BB1, St, R2, BL]. We also deal with superlinear Sturm—Liouville problem: in (0,1) and give a complete characterization of the solutions related to the Symmetric Mountain Pass Theorem

Homoclinic orbits in a first order superquadratic hamiltonian system: Convergence of subharmonic orbits

Abstract We consider the existence of homoclinic orbits for a first order Hamiltonian system z = JH z (t, z) . We assume H(t, z) is of form H(t, z) = 1 2 (Az, z) + W(t, z) , where A is a symmetric matrix with δ(JA)∩i R = ∅ and W(t, z) is 2π-periodic in t and has superquadratic growth in z. We prove the existence of a nontrivial homoclinic solution z∞(t) and subharmonic solutions (zT(t))TϵN (i.e., 2πT-periodic solutions) of (HS) such that ZT(t) → Z∞(t) in Cloc1( R , R 2N) as T → ∞.

Homoclinic orbits for a singular second order Hamiltonian system

Abstract We consider the second order Hamiltonian system: (HS) q .. + V ′ ( q ) = 0 where q = (q1, …, qN) ∈ RN(N ≧ 3) and V:RN\{e} → R(e ∈ RN) is a potential with a singularity, i.e., |V(q)| → ∞ as q →e. We prove the existence of a homoclinic orbit of (HS) under suitable assumptions. Our main assumptions are the strong force condition of Gordon [8] and the uniqueness of a global maximum of V.

Standing Waves for Nonlinear Schrödinger Equations with a General Nonlinearity: One and Two Dimensional Cases

For N = 1,2, we consider singularly perturbed elliptic equations ϵ2Δ u − V(x) u + f(u)= 0, u(x)> 0 on R N , lim|x|→∞ u(x)= 0. For small ϵ > 0, we show the existence of a localized bound state solution concentrating at an isolated component of positive local minimum of V under conditions on f we believe to be almost optimal; when N ≥ 3, it was shown in Byeon and Jeanjean (2007).

Semi-classical standing waves for nonlinear Schrödinger equations at structurally stable critical points of the potential

We consider a singularly perturbed elliptic equation ε21u− V (x)u+ f (u) = 0, u(x) > 0 on R , lim |x|→∞ u(x) = 0, where V (x) > 0 for any x ∈ RN . The singularly perturbed problem has corresponding limiting problems 1U − cU + f (U) = 0, U(x) > 0 on R , lim |x|→∞ U(x) = 0, c > 0. Berestycki–Lions [3] found almost necessary and sufficient conditions on the nonlinearity f for existence of a solution of the limiting problem. There have been endeavors to construct solutions of the singularly perturbed problem concentrating around structurally stable critical points of the potential V under possibly general conditions on f . In this paper, we prove that under the optimal conditions of Berestycki–Lions on f ∈ C1, there exists a solution concentrating around topologically stable positive critical points of V , whose critical values are characterized by minimax methods.

A note on a mountain pass characterization of least energy solutions

Abstract We consider the equation -uʺ = g(u), u(x) ∈ H1(ℝ). (0.1) Under general assumptions on the nonlinearity g we prove that the, unique up to translation, solution of (0.1) is at the mountain pass level of the associated functional. This result extends a corresponding result for least energy solutions when (0.1) is set on ℝN.

Non-collision solutions for a second order singular Hamiltonian system with weak force

Abstract Under a weak force type condition, we consider the existence of time periodic solutions of singular Hamiltonian systems: (HS) q ¨ + V q ( q , t ) = 0 q ( t + T ) = q ( t ) . } We assume V (q, t) V ( q , t ) = − 1 | q | α + U ( q , t ) where 0 For α ∈ (1, 2], we prove the existence of a non-collision solution of (HS). For α ∈ (0, 1], we prove that the generalized solution of (HS), which is introduced in [BR], enters the singularity 0 at most one time in its period. Our argument depends on a minimax argument due to [BR] and an estimate of Morse index of corresponding functional, which will be obtained via re-scaling argument.