Abstract
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: d d t ( | u ˙ ( t ) | p − 2 u ˙ ( t ) ) − a ( t ) | u ( t ) | q − p u ( t ) + ∇ W ( t , u ( t ) ) = 0 , where 1 < p < ( q + 2 ) / 2 , q > 2 , t ∈ R , u ∈ R N , a ∈ C ( R , R ) and W ∈ C 1 ( R × R N , R ) are not periodic in t. MSC:34C37, 35A15, 37J45, 47J30.
Highlights
Consider homoclinic solutions of the following p-Laplacian system:d u (t) p– u (t) – a(t) u(t) q–pu(t) + ∇W t, u(t) =, t ∈ R, ( . )dt where < p < (q + )/, q >, t ∈ R, u ∈ RN, a : R → R, W : R × RN → R
We say that a solution u of ( . ) is a nontrivial homoclinic if u ∈ C (R, RN ) such that u =, u(t) → as t → ±∞
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 Poincaré [ ]
Summary
). In [ ], by introducing a suitable Sobolev space, Salvatore established the following existence results for system 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 When W (t, x) is an even function in x, Salvatore [ ] obtained the following existence theorem of an unbounded sequence of homoclinic orbits for problem
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