Solutions of a system of diffusion equations

Abstract

×ØÖ Øº Ï ×ØÙ Ý Ü ×Ø Ò Ò ÑÙÐØ ÔÐ ØÝ Ó ÓÑÓ Ð Ò ØÝÔ ×ÓÐÙØ ÓÒ× Hv(t, x, u, v), t, x, u, v),x))¸V ∈ C(Ω, R) Ò H ∈ C 1 (R × Ω × R 2m , R)º ËÙ ÔÖÓ Ð Ñ× Ö × Ò ÓÔØ Ñ Ð ÓÒØÖÓÐ Ó ×Ý×Ø Ñ×

Homoclinic orbits for an unbounded superquadratic

“…Proof. If B(t, x) ≡ 0, the proof was given in [4,16]. Therefore, in our case, the matrix B(t, x) = 0 is bounded.…”

“…Barbu [7]), or an infinite-dimensional Hamiltonian system in L 2 (R × R N , R 2m ) (cf. [4,16,17]). In this paper, we are interested in the case that…”

“…Nearly, Ding et al [16] considered (HS) with b(t, x) ≡ 0 via variational methods. Under some periodic assumptions on V (x), b(t, x) and G(t, x, z), the existence of infinitely many solutions were obtained in their paper [16] for both superquadratic or asymptotically linear cases. For nonperiodic case, there has not much works done up to know.…”

“…Note that, by (H 0 )-(H 4 ), it is not difficult to show that S min is bounded in E (cf. [16] or [18]), hence, |u| 2 ≤ C 2 for all u ∈ S min and some C 2 . Therefore, as a consequence of Lemma 2.8 we see that, for each q ∈ [2, ∞), there is Λ q > 0 such that…”

On a class of infinite-dimensional Hamiltonian systems with asymptotically periodic nonlinearities

“…In the case b(t, x) ≡ 0, V (x) = 0, Bartsch and Ding [4] investigated the following infinitedimensional Hamiltonian system: For the case b(t, x) = 0, V (x) = 0, the diffusion equations with periodic potential and nonlinearities was recently considered by Ding, Luan and Willem in [14]. They assumed that H(t, x, 0) ≡ 0 and H(t, x, z) is asymptotically quadratic or super-quadratic as |z| → ∞.…”