Existence of homoclinic orbits for a class of p -Laplacian systems in a weighted Sobolev space

  • PDF / 265,133 Bytes
  • 18 Pages / 595.28 x 793.7 pts Page_size
  • 87 Downloads / 232 Views

DOWNLOAD

REPORT


RESEARCH

Open Access

Existence of homoclinic orbits for a class of p-Laplacian systems in a weighted Sobolev space Xiubo Shi1 , Qiongfen Zhang1* and Qi-Ming Zhang2 *

Correspondence: [email protected] 1 College of Science, Guilin University of Technology, Guilin, Guangxi 541004, P.R. China Full list of author information is available at the end of the article

Abstract By applying the mountain pass theorem and symmetric mountain pass theorem in critical point theory, the existence of at least one or infinitely many homoclinic solutions is obtained for the following p-Laplacian system:  q–p d  p–2 ( u˙ (t) u˙ (t)) – a(t)u(t) u(t) + ∇W (t, u(t)) = 0, dt where 1 < p < (q + 2)/2, q > 2, t ∈ R, u ∈ RN , a ∈ C(R, R) and W ∈ C 1 (R × RN , R) are not periodic in t. MSC: 34C37; 35A15; 37J45; 47J30 Keywords: homoclinic solutions; variational methods; weighted Lq–p+2 space; p-Laplacian systems

1 Introduction Consider homoclinic solutions of the following p-Laplacian system:  q–p p–    d  ˙ ˙  u(t) – a(t)u(t) u(t) + ∇W t, u(t) = , u(t) dt

t ∈ R,

(.)

where  < p < (q + )/, q > , t ∈ R, u ∈ RN , a : R → R, W : R × RN → R. As usual, we say that a solution u of (.) is a nontrivial homoclinic (to ) if u ∈ C  (R, RN ) such that u = , u(t) →  as t → ±∞. When p = , (.) reduces to the following second-order Hamiltonian system:  q–   ¨ – a(t)u(t) u(t) + ∇W t, u(t) = , u(t)

t ∈ R.

(.)

If we take p =  and q = , then (.) reduces to the following second-order Hamiltonian system:   ¨ – a(t)u(t) + ∇W t, u(t) = , u(t)

t ∈ R.

(.)

The existence of homoclinic orbits for Hamiltonian systems is a classical problem and its importance in the study of the behavior of dynamical systems has been recognized by © 2013 Shi et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Shi et al. Boundary Value Problems 2013, 2013:137 http://www.boundaryvalueproblems.com/content/2013/1/137

Page 2 of 18

Poincaré []. Up to the year of , a few of isolated results can be found, and the only method for dealing with such a problem was the small perturbation technique of Melnikov. Recently, the existence and multiplicity of homoclinic solutions and periodic solutions for Hamiltonian systems have been extensively studied by critical point theory. For example, see [–] and references therein. However, few results [, ] have been obtained in the literature for system (.). In [], by introducing a suitable Sobolev space, Salvatore established the following existence results for system (.) when q > . Theorem A [] Assume that a and W satisfy the following conditions: (A)

Let q > , a(t) is a continuous, positive function on R such that for all t ∈ R a(t) ≥ α |t|β ,

α > , β > (q – )/.

(W) W ∈ C  (R × RN , R) and there exists a constant μ