Open Access

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

Boundary Value Problems20132013:137

DOI: 10.1186/1687-2770-2013-137

Received: 5 December 2012

Accepted: 7 May 2013

Published: 24 May 2013

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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equa_HTML.gif

where 1 < p < ( q + 2 ) / 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq1_HTML.gif, q > 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq2_HTML.gif, t R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq3_HTML.gif, u R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq4_HTML.gif, a C ( R , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq5_HTML.gif and W C 1 ( R × R N , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq6_HTML.gif are not periodic in t.

MSC:34C37, 35A15, 37J45, 47J30.

Keywords

homoclinic solutions variational methods weighted L q p + 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq7_HTML.gif space p-Laplacian systems

1 Introduction

Consider homoclinic solutions of 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 , t R , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ1_HTML.gif
(1.1)

where 1 < p < ( q + 2 ) / 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq1_HTML.gif, q > 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq2_HTML.gif, t R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq8_HTML.gif, u R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq9_HTML.gif, a : R R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq10_HTML.gif, W : R × R N R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq11_HTML.gif. As usual, we say that a solution u of (1.1) is a nontrivial homoclinic (to 0) if u C 2 ( R , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq12_HTML.gif such that u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq13_HTML.gif, u ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq14_HTML.gif as t ± https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq15_HTML.gif.

When p = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq16_HTML.gif, (1.1) reduces to the following second-order Hamiltonian system:
u ¨ ( t ) a ( t ) | u ( t ) | q 2 u ( t ) + W ( t , u ( t ) ) = 0 , t R . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ2_HTML.gif
(1.2)
If we take p = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq16_HTML.gif and q = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq17_HTML.gif, then (1.2) reduces to the following second-order Hamiltonian system:
u ¨ ( t ) a ( t ) u ( t ) + W ( t , u ( t ) ) = 0 , t R . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ3_HTML.gif
(1.3)

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 [1]. Up to the year of 1990, 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 [220] and references therein. However, few results [21, 22] have been obtained in the literature for system (1.2). In [22], by introducing a suitable Sobolev space, Salvatore established the following existence results for system (1.2) when q > 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq2_HTML.gif.

Theorem A [22]

Assume that a and W satisfy the following conditions:
  1. (A)
    Let q > 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq2_HTML.gif, a ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq18_HTML.gif is a continuous, positive function on such that for all t R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq3_HTML.gif
    a ( t ) α 0 | t | β 0 , α 0 > 0 , β 0 > ( q 2 ) / 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equb_HTML.gif
     
(W1) W C 1 ( R × R N , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq19_HTML.gif and there exists a constant μ > q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq20_HTML.gif such that
0 < μ W ( t , x ) ( W ( t , x ) , x ) , ( t , x ) R × R N { 0 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equc_HTML.gif

(W2) | W ( t , x ) | = o ( | x | q 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq21_HTML.gif as | x | 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq22_HTML.gif uniformly with respect to t R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq8_HTML.gif.

(W3) There exists W ¯ C ( R N , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq23_HTML.gif such that
| W ( t , x ) | + | W ( t , x ) | | W ¯ ( x ) | , ( t , x ) R × R N . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equd_HTML.gif

Then problem (1.2) has one nontrivial homoclinic solution.

When W ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq24_HTML.gif is an even function in x, Salvatore [22] obtained the following existence theorem of an unbounded sequence of homoclinic orbits for problem (1.2) by the symmetric mountain pass theorem.

Theorem B [22]

Assume that a and W satisfy (A), (W1)-(W3) and the following condition:

(W4) W ( t , x ) = W ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq25_HTML.gif, ( t , x ) R × R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq26_HTML.gif.

Then problem (1.2) has an unbounded sequence of homoclinic solutions.

In [21], Chen and Tang improved Theorem A and Theorem B by relaxing conditions (W1) and (W2) and removing condition (W3). Motivated mainly by the ideas of [18, 2123], we will consider homoclinic solutions of (1.1) by the mountain pass theorem and symmetric mountain pass theorem. Precisely, we obtain the following main results.

Theorem 1.1 Suppose that a and W satisfy the following conditions:

(A)′ Let 1 < p < ( q + 2 ) / 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq1_HTML.gif and q > 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq2_HTML.gif, a ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq18_HTML.gif is a continuous, positive function on such that for all t R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq3_HTML.gif
a ( t ) α | t | β , α > 0 , β > ( q 2 p + 2 ) / p . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Eque_HTML.gif
(W5) W ( t , x ) = W 1 ( t , x ) W 2 ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq27_HTML.gif, W 1 , W 2 C 1 ( R × R N , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq28_HTML.gif, and there exists a constant R > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq29_HTML.gif such that
1 a ( t ) | W ( t , x ) | = o ( | x | q p + 1 ) as x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equf_HTML.gif

uniformly in t ( , R ] [ R , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq30_HTML.gif.

(W6) There is a constant μ > q p + 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq31_HTML.gif such that
0 < μ W 1 ( t , x ) ( W 1 ( t , x ) , x ) , ( t , x ) R × R N { 0 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equg_HTML.gif
(W7) W 2 ( t , 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq32_HTML.gif and there exists a constant ϱ ( q p + 2 , μ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq33_HTML.gif such that
W 2 ( t , x ) 0 , ( W 2 ( t , x ) , x ) ϱ W 2 ( t , x ) , ( t , x ) R × R N . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equh_HTML.gif

Then problem (1.1) has one nontrivial homoclinic solution.

Theorem 1.2 Suppose that a and W satisfy (A)′, (W6) and the following conditions:

(W5)′ W ( t , x ) = W 1 ( t , x ) W 2 ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq27_HTML.gif, W 1 , W 2 C 1 ( R × R N , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq28_HTML.gif, and
1 a ( t ) | W ( t , x ) | = o ( | x | q p + 1 ) as x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equi_HTML.gif

uniformly in t R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq3_HTML.gif.

(W7)′ W 2 ( t , 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq32_HTML.gif and there exists a constant ϱ ( q p + 2 , μ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq33_HTML.gif such that
( W 2 ( t , x ) , x ) ϱ W 2 ( t , x ) , ( t , x ) R × R N . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equj_HTML.gif

Then problem (1.1) has one nontrivial homoclinic solution.

Theorem 1.3 Suppose that a and W satisfy (A)′ and (W4)-(W7). Then problem (1.1) has an unbounded sequence of homoclinic solutions.

Theorem 1.4 Suppose that a and W satisfy (A)′, (W4), (W5)′, (W6), (W7)′. Then problem (1.1) has an unbounded sequence of homoclinic solutions.

Remark 1.1 When p = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq16_HTML.gif, condition (A)′ reduces to condition (A). Obviously, Theorem 1.1-Theorem 1.4 generalize and improve Theorem A, Theorem B and the corresponding results in [21]. It is easy to see that our results hold true even if p = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq16_HTML.gif. To the best of our knowledge, similar results for problem (1.1) cannot be seen in the literature; from this point, our results are new.

Remark 1.2 If p = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq16_HTML.gif and q = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq17_HTML.gif, then problem (1.1) reduces to problem (1.3). As pointed out in [23], Theorem A can be proved by replacing (A) with the more general assumption: a ( t ) + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq34_HTML.gif as | t | + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq35_HTML.gif.

The rest of this paper is organized as follows. In Section 2, some preliminaries are presented and we establish an embedding result. In Section 3, we give the proofs of our results. In Section 4, some examples are given to illustrate our results.

2 Preliminaries

We set, for any real number 1 h < + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq36_HTML.gif,
L h = L h ( R , R N ) , L = L ( R , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equk_HTML.gif
with the usual norms
u h = ( R | u ( t ) | h d t ) 1 / h , u = max t R | u ( t ) | . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equl_HTML.gif
Let
W 1 , p = W 1 , p ( R , R N ) = { u : R R N u  is absolutely continuous , u , u ˙ L p ( R , R N ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equm_HTML.gif
be the Sobolev space with the norm given by
u W 1 , p = ( R [ | u ˙ ( t ) | p + | u ( t ) | p ] d t ) 1 / p . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equn_HTML.gif
If σ is a positive, continuous function on and 1 < s < + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq37_HTML.gif, let
L σ s = L σ s ( R , R N ; σ ) = { u L loc 1 ( R , R N ) | R σ ( t ) | u ( t ) | s d t < + } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equo_HTML.gif
L σ s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq38_HTML.gif equipped with the norm
u s , σ = ( R σ ( t ) | u ( t ) | s d t ) 1 / s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equp_HTML.gif
is a reflexive Banach space. When s = + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq39_HTML.gif, we set
L σ = L σ ( R , R N ; σ ) = { u | max t R σ ( t ) | u ( t ) | < + } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equq_HTML.gif
with the norm given by
u , σ = max t R σ ( t ) | u ( t ) | . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equr_HTML.gif
Set E = W 1 , p L a q p + 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq40_HTML.gif, where a is the function given in condition (A)′. Then E with its standard norm https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq41_HTML.gif is a reflexive Banach space. The functional φ corresponding to (1.1) on E is given by
φ ( u ) = R [ 1 p | u ˙ ( t ) | p + a ( t ) q p + 2 | u ( t ) | q p + 2 W ( t , u ( t ) ) ] d t , u E . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ4_HTML.gif
(2.1)
Clearly, it follows from (W5) or (W5)′ that φ : E R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq42_HTML.gif. By Theorem 2.1 of [24], we can deduce that the map
u a ( t ) | u ( t ) | q p u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equs_HTML.gif
is continuous from L a q p + 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq43_HTML.gif in the dual space L a 1 / ( q p + 1 ) p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq44_HTML.gif, where p 1 = q p + 2 q p + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq45_HTML.gif. As the embeddings E W 1 , p L γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq46_HTML.gif for all γ p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq47_HTML.gif are continuous, if (A)′ and (W5) or (W5)′ hold, then φ C 1 ( E , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq48_HTML.gif and one can easily check that
φ ( u ) , v = R [ | u ˙ ( t ) | p 2 ( u ˙ ( t ) , v ˙ ( t ) ) + a ( t ) | u ( t ) | q p ( u ( t ) , v ( t ) ) ( W ( t , u ( t ) ) , v ( t ) ) ] d t , u E . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ5_HTML.gif
(2.2)

Furthermore, the critical points of φ in E are classical solutions of (1.1) with u ( ± ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq49_HTML.gif.

To prove our results, we need the following generalization of the Lebesgue dominated convergence theorem.

Lemma 2.1 [25]

Let { f n ( t ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq50_HTML.gif and { g n ( t ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq51_HTML.gif be two sequences of measurable functions on a measurable set A, and let
| f n ( t ) | g n ( t ) for a.e. t A . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equt_HTML.gif
If
lim n f n ( t ) = f ( t ) , lim n g n ( t ) = g ( t ) for a.e. t A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equu_HTML.gif
and
lim n A g n ( t ) d t = A g ( t ) d t < + , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equv_HTML.gif
then
lim n A f n ( t ) d t = A f ( t ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equw_HTML.gif

The following lemma is an improvement result of [23] in which the author considered the case p = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq16_HTML.gif.

Lemma 2.2 If a satisfies assumption (A)′, then
the embedding L a q p + 2 L p is continuous . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ6_HTML.gif
(2.3)
Moreover, there exists a Sobolev space Z such that
the embeddings L a q p + 2 Z L p are continuous , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ7_HTML.gif
(2.4)
the embedding W 1 , p Z L p is compact . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ8_HTML.gif
(2.5)
Proof Let θ = ( q p + 2 ) / ( q 2 p + 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq52_HTML.gif, θ = ( q p + 2 ) / p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq53_HTML.gif, we have
u p p = R a 1 / θ a 1 / θ | u | p d t ( R a θ / θ d t ) 1 / θ ( R a | u | p θ d t ) 1 / θ = a 1 ( R a | u | q p + 2 d t ) p / q p + 2 = a 1 u q p + 2 , a p , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equx_HTML.gif

where from (A)′, a 1 = ( R a p / ( q 2 p + 2 ) d t ) ( q 2 p + 2 ) / ( q p + 2 ) < + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq54_HTML.gif. Then (2.3) holds.

By (A)′, there exists a continuous positive function ρ such that ρ ( t ) + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq55_HTML.gif as | t | + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq35_HTML.gif and
a 2 = ( R ρ θ a θ / θ d t ) 1 / θ < + . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equy_HTML.gif
Since
u p , ρ p = R ρ | u | p d t = R ρ a 1 / θ a 1 / θ | u | p d t ( R ρ θ a θ / θ d t ) 1 / θ ( R a | u | q p + 2 d t ) 1 / θ = a 2 u q p + 2 , a p , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equz_HTML.gif

(2.4) holds by taking Z = L ρ p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq56_HTML.gif.

Finally, as W 1 , p Z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq57_HTML.gif is the weighted Sobolev space Γ 1 , p ( R , ρ , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq58_HTML.gif, it follows from [24] that (2.5) holds. □

The following two lemmas are the mountain pass theorem and symmetric mountain pass theorem, which are useful in the proofs of our theorems.

Lemma 2.3 [26]

Let E be a real Banach space and I C 1 ( E , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq59_HTML.gif satisfying (PS)-condition. Suppose I ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq60_HTML.gif and:
  1. (i)

    There exist constants ρ , α > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq61_HTML.gif such that I B ρ ( 0 ) α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq62_HTML.gif.

     
  2. (ii)

    There exists an e E B ¯ ρ ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq63_HTML.gif such that I ( e ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq64_HTML.gif.

     
Then I possesses a critical value c α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq65_HTML.gif which can be characterized as
c = inf h Φ max s [ 0 , 1 ] I ( h ( s ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equaa_HTML.gif

where Φ = { h C ( [ 0 , 1 ] , E ) h ( 0 ) = 0 , h ( 1 ) = e } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq66_HTML.gif and B ρ ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq67_HTML.gif is an open ball in E of radius ρ centered at 0.

Lemma 2.4 [26]

Let E be a real Banach space and I C 1 ( E , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq59_HTML.gif with I even. Assume that I ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq60_HTML.gif and I satisfies (PS)-condition, (i) of Lemma  2.3 and the following condition:
  1. (iii)

    For each finite dimensional subspace E E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq68_HTML.gif, there is r = r ( E ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq69_HTML.gif such that I ( u ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq70_HTML.gif for u E B r ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq71_HTML.gif, B r ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq72_HTML.gif is an open ball in E of radius r centered at 0.

     

Then I possesses an unbounded sequence of critical values.

Lemma 2.5 Assume that (W6) and (W7) or (W7)′ hold. Then, for every ( t , x ) R × R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq73_HTML.gif,
  1. (i)

    s μ W 1 ( t , s x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq74_HTML.gif is nondecreasing on ( 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq75_HTML.gif;

     
  2. (ii)

    s ϱ W 2 ( t , s x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq76_HTML.gif is nonincreasing on ( 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq75_HTML.gif.

     

The proof of Lemma 2.5 is routine and we omit it.

3 Proofs of theorems

Proof of Theorem 1.1 Step 1. The functional φ satisfies (PS)-condition. Let { u n } E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq77_HTML.gif satisfying φ ( u n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq78_HTML.gif be bounded and φ ( u n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq79_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq80_HTML.gif. Hence, there exists a constant C 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq81_HTML.gif such that
| φ ( u n ) | C 1 , φ ( u n ) E μ C 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ9_HTML.gif
(3.1)
It is well known [27] that there exists a constant C 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq82_HTML.gif such that
u C 2 u , u E . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ10_HTML.gif
(3.2)
From (2.1), (2.2), (3.1), (W6) and (W7), we have
p C 1 + p C 1 u n p φ ( u n ) p μ φ ( u n ) , u n = μ p μ u ˙ n p p + p R [ W 2 ( t , u n ( t ) ) 1 μ ( W 2 ( t , u n ( t ) ) , u n ( t ) ) ] d t p R [ W 1 ( t , u n ( t ) ) 1 μ ( W 1 ( t , u n ( t ) ) , u n ( t ) ) ] d t + ( p q p + 2 p μ ) R a ( t ) | u n ( t ) | q p + 2 d t μ p μ u ˙ n p p + ( p q p + 2 p μ ) u n q p + 2 , a q p + 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equab_HTML.gif
It follows from Lemma 2.2, p < ( q + 2 ) / 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq83_HTML.gif, μ > q p + 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq31_HTML.gif and the above inequalities that there exists a constant C 3 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq84_HTML.gif such that
u n C 3 , n N . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ11_HTML.gif
(3.3)
Now we prove that u n u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq85_HTML.gif in E. Passing to a subsequence if necessary, it can be assumed that u n u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq86_HTML.gif in E. From Lemma 2.2, we have u n u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq85_HTML.gif in L p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq87_HTML.gif. From (3.2) and (3.3), we have
u n C 2 u n C 2 C 3 , u n E . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ12_HTML.gif
(3.4)
Inequality (3.4) implies that | u n ( t ) | C 2 C 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq88_HTML.gif for all t R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq8_HTML.gif. By (W5), we know that
| W ( t , x ) | a ( t ) | x | q p + 1 0 as  x 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equac_HTML.gif
which implies that for any given constant C > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq89_HTML.gif, there exists a constant C > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq90_HTML.gif related to C such that
| W ( t , x ) | a ( t ) | x | q p + 1 C for  | x | C . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equad_HTML.gif
Hence, there exists a constant C 4 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq91_HTML.gif such that
| W ( t , x ) | C 4 a ( t ) | x | q p + 1 for  | x | C 2 C 3 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ13_HTML.gif
(3.5)
Hence, from (3.5), we have
| W ( t , u n ( t ) ) W ( t , u 0 ( t ) ) | p [ C 4 a ( t ) ( | u n ( t ) | q p + 1 + | u 0 ( t ) | q p + 1 ) ] p [ C 4 2 q p + 1 a ( t ) | u n ( t ) u 0 ( t ) | q p + 1 + C 4 ( 1 + 2 q p + 1 ) a ( t ) | u 0 ( t ) | q p + 1 ] p 2 p ( q p + 2 ) C 4 p a p ( t ) | u n ( t ) u 0 ( t ) | p ( q p + 1 ) + 2 p C 4 p ( 1 + 2 q p + 1 ) p a p ( t ) | u 0 ( t ) | p ( q p + 1 ) : = g n ( t ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ14_HTML.gif
(3.6)
where p = p p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq92_HTML.gif. Moreover, since a ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq18_HTML.gif is a positive continuous function on , p < q p + 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq93_HTML.gif and u n ( t ) u 0 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq94_HTML.gif for almost every t R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq8_HTML.gif, we have
lim n g n ( t ) = 2 p C 4 p ( 1 + 2 q p + 1 ) p a p ( t ) | u 0 ( t ) | p ( q p + 1 ) : = g ( t ) for a.e.  t R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equae_HTML.gif
and
lim n R g n ( t ) d t = lim n R [ 2 p ( q p + 2 ) C 4 p a p ( t ) | u n ( t ) u 0 ( t ) | p ( q p + 1 ) + 2 p C 4 p ( 1 + 2 q p + 1 ) p a p ( t ) | u 0 ( t ) | p ( q p + 1 ) ] d t = 2 p ( q p + 2 ) C 4 p lim n R a p ( t ) | u n ( t ) u 0 ( t ) | p ( q p + 1 ) d t + 2 p C 4 p ( 1 + 2 q p + 1 ) p R a p ( t ) | u 0 ( t ) | p ( q p + 1 ) d t = 2 p C 4 p ( 1 + 2 q p + 1 ) p R a p ( t ) | u 0 ( t ) | p ( q p + 1 ) d t = R g ( t ) d t < + . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equaf_HTML.gif
It follows from Lemma 2.1, (3.6) and the above inequalities that
lim n R | W ( t , u n ( t ) ) W ( t , u 0 ( t ) ) | p d t = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equag_HTML.gif
This shows that
W ( t , u n ) W ( t , u 0 ) in  L p ( R , R N ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ15_HTML.gif
(3.7)
From (2.2), we have
φ ( u n ) φ ( u 0 ) , u n u 0 ) = R ( | u ˙ n ( t ) | p 2 u ˙ n ( t ) | u ˙ 0 ( t ) | p 2 u ˙ 0 ( t ) , u ˙ n ( t ) u ˙ 0 ( t ) ) d t + R a ( t ) ( | u n ( t ) | q p u n ( t ) | u 0 ( t ) | q p u 0 ( t ) ) ( u n ( t ) u 0 ( t ) ) d t R ( W ( t , u n ( t ) ) W ( t , u 0 ( t ) ) , u n ( t ) u 0 ( t ) ) d t u ˙ n p p + u ˙ 0 p p u ˙ 0 p u ˙ n p p 1 u ˙ n p u ˙ 0 p p 1 + R a ( t ) ( | u n ( t ) | q p u n ( t ) | u 0 ( t ) | q p u 0 ( t ) ) ( u n ( t ) u 0 ( t ) ) d t R ( W ( t , u n ( t ) ) W ( t , u 0 ( t ) ) , u n ( t ) u 0 ( t ) ) d t = ( u ˙ n p p 1 u ˙ 0 p p 1 ) ( u ˙ n p u ˙ 0 p ) + R a ( t ) ( | u n ( t ) | q p u n ( t ) | u 0 ( t ) | q p u 0 ( t ) ) ( u n ( t ) u 0 ( t ) ) d t R ( W ( t , u n ( t ) ) W ( t , u 0 ( t ) ) , u n ( t ) u 0 ( t ) ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ16_HTML.gif
(3.8)
It is easy to see that for any α > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq95_HTML.gif there exists a constant C 5 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq96_HTML.gif such that
( | x | α 1 x | y | α 1 y ) ( x y ) C 5 | x y | α + 1 , x , y R . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ17_HTML.gif
(3.9)
Inequality (3.9) implies that
( u ˙ n p p 1 u ˙ 0 p p 1 ) ( u ˙ n p u ˙ 0 p ) C 5 | u ˙ n p u ˙ 0 p | p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ18_HTML.gif
(3.10)
and
R a ( t ) ( | u n ( t ) | q p u n ( t ) | u 0 ( t ) | q p u 0 ( t ) ) ( u n ( t ) u 0 ( t ) ) d t C 5 R a ( t ) | u n ( t ) u 0 ( t ) | q p + 2 d t , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ19_HTML.gif
(3.11)
where C 5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq97_HTML.gif and C 5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq98_HTML.gif are positive constants. Since φ ( u n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq79_HTML.gif as n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq99_HTML.gif, u n u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq100_HTML.gif in E and the embeddings E W 1 , p L γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq46_HTML.gif for all γ p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq47_HTML.gif are continuous, it follows from Lemma 2.2, (3.7), (3.8), (3.10) and (3.11) that
u ˙ n p u ˙ 0 p as  n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ20_HTML.gif
(3.12)
and
R a ( t ) | u n ( t ) | q p + 2 d t R a ( t ) | u 0 ( t ) | q p + 2 d t as  n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ21_HTML.gif
(3.13)

Hence, by (3.12) and (3.13), u n u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq85_HTML.gif in E. This shows that φ satisfies (PS)-condition.

Step 2. From (W5), there exists δ ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq101_HTML.gif such that
| W ( t , x ) | 1 p a ( t ) | x | q p + 1 for  | t | R , | x | δ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ22_HTML.gif
(3.14)
By (3.14), we have
| W ( t , x ) | 1 p ( q p + 2 ) a ( t ) | x | q p + 2 for  | t | R , | x | δ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ23_HTML.gif
(3.15)
Let
C 6 = sup { W 1 ( t , x ) a ( t ) | t [ R , R ] , x R , | x | = 1 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ24_HTML.gif
(3.16)
Set σ = min { 1 / ( p ( q p + 2 ) C 6 + 1 ) 1 / ( μ q + p 2 ) , δ } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq102_HTML.gif and u = σ / C 2 : = ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq103_HTML.gif, it follows from (3.2) that
u C 2 u σ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equah_HTML.gif
which shows that | u ( t ) | σ δ < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq104_HTML.gif. From Lemma 2.5(i) and (3.16), we have
R R W 1 ( t , u ( t ) ) d t { t [ R , R ] : u ( t ) 0 } W 1 ( t , u ( t ) | u ( t ) | ) | u ( t ) | μ d t C 6 R R a ( t ) | u ( t ) | μ d t C 6 σ μ q + p 2 R R a ( t ) | u ( t ) | q p + 2 d t 1 p ( q p + 2 ) R R a ( t ) | u ( t ) | q p + 2 d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ25_HTML.gif
(3.17)
It follows from (W7), (3.15), (3.17) that
φ ( u ) = 1 p R | u ˙ ( t ) | p d t + R a ( t ) q p + 2 | u ( t ) | q p + 2 d t R W ( t , u ( t ) ) d t = 1 p u ˙ p p + 1 q p + 2 u q p + 2 , a q p + 2 R [ R , R ] W ( t , u ( t ) ) d t R R W ( t , u ( t ) ) d t 1 p u ˙ p p + 1 q p + 2 u q p + 2 , a q p + 2 R R W 1 ( t , u ( t ) ) d t R [ R , R ] 1 p ( q p + 2 ) a ( t ) | u ( t ) | q p + 2 d t 1 p u ˙ p p + 1 q p + 2 u q p + 2 , a q p + 2 1 p ( q p + 2 ) R R a ( t ) | u ( t ) | q p + 2 d t R [ R , R ] 1 p ( q p + 2 ) a ( t ) | u ( t ) | q p + 2 d t = 1 p u ˙ p p + p 1 p ( q p + 2 ) u q p + 2 , a q p + 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equai_HTML.gif

Therefore, we can choose a constant α > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq105_HTML.gif depending on ρ such that φ ( u ) α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq106_HTML.gif for any u E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq107_HTML.gif with u = ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq108_HTML.gif.

Step 3. From Lemma 2.5(ii) and (3.2), we have for any u E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq107_HTML.gif
2 2 W 2 ( t , u ( t ) ) d t = { t [ 2 , 2 ] : | u ( t ) | > 1 } W 2 ( t , u ( t ) ) d t + { t [ 2 , 2 ] : | u ( t ) | 1 } W 2 ( t , u ( t ) ) d t { t [ 2 , 2 ] : | u ( t ) | > 1 } W 2 ( t , u ( t ) | u ( t ) | ) | u ( t ) | ϱ d t + 2 2 max | x | 1 W 2 ( t , x ) d t u ϱ 2 2 max | x | = 1 W 2 ( t , x ) d t + 2 2 max | x | 1 W 2 ( t , x ) d t C 2 ϱ u ϱ 2 2 max | x | = 1 W 2 ( t , x ) d t + 2 2 max | x | 1 W 2 ( t , x ) d t = C 7 u ϱ + C 8 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ26_HTML.gif
(3.18)
where C 7 = C 2 ϱ 2 2 max | x | = 1 W 2 ( t , x ) d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq109_HTML.gif, C 8 = 2 2 max | x | 1 W 2 ( t , x ) d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq110_HTML.gif. Take ω E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq111_HTML.gif such that
| ω ( t ) | = { 1 for  | t | 1 , 0 for  | t | 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ27_HTML.gif
(3.19)
and | ω ( t ) | 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq112_HTML.gif for | t | ( 1 , 2 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq113_HTML.gif. For s > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq114_HTML.gif, from Lemma 2.5(i) and (3.19), we get
1 1 W 1 ( t , s ω ( t ) ) d t s μ 1 1 W 1 ( t , ω ( t ) ) d t = C 9 s μ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ28_HTML.gif
(3.20)
where C 9 = 1 1 W 1 ( t , ω ( t ) ) d t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq115_HTML.gif. From (W7), (2.1), (3.18), (3.19), (3.20), we get for s > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq114_HTML.gif
φ ( s ω ) = s p p ω ˙ p p + s q p + 2 q p + 2 ω q p + 2 , a q p + 2 + R [ W 2 ( t , s ω ( t ) ) W 1 ( t , s ω ( t ) ) ] d t s p p ω ˙ p p + s q p + 2 q p + 2 ω q p + 2 , a q p + 2 + 2 2 W 2 ( t , s ω ( t ) ) d t 1 1 W 1 ( t , s ω ( t ) ) d t s p p ω ˙ p p + s q p + 2 q p + 2 ω q p + 2 , a q p + 2 + C 7 s ϱ ω ϱ + C 8 C 9 s μ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ29_HTML.gif
(3.21)
Since μ > ϱ > q p + 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq116_HTML.gif and C 9 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq117_HTML.gif, it follows from (3.21) that there exists s 1 > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq118_HTML.gif such that s 1 ω > ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq119_HTML.gif and φ ( s 1 ω ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq120_HTML.gif. Let e = s 1 ω ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq121_HTML.gif, then e E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq122_HTML.gif, e = s 1 ω > ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq123_HTML.gif and φ ( e ) = φ ( s 1 ω ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq124_HTML.gif. By Lemma 2.3, φ has a critical value d > α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq125_HTML.gif given by
d = inf g Φ max s [ 0 , 1 ] φ ( g ( s ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ30_HTML.gif
(3.22)
where
Φ = { g C ( [ 0 , 1 ] , E ) : g ( 0 ) = 0 , g ( 1 ) = e } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equaj_HTML.gif
Hence, there exists u E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq126_HTML.gif such that
φ ( u ) = d , φ ( u ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equak_HTML.gif

The function u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq127_HTML.gif is a desired solution of problem (1.1). Since d > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq128_HTML.gif, u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq127_HTML.gif is a nontrivial homoclinic solution. The proof is complete. □

Proof of Theorem 1.2 In the proof of Theorem 1.1, the condition W 2 ( t , x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq129_HTML.gif in (W7) is only used in the proofs of (3.3) and Step 2. Therefore, we only need to prove that (3.3) and Step 2 still hold if we use (W5)′ and (W7)′ instead of (W5) and (W7). We first prove that (3.3) holds. From (W6), (W7)′, (2.1), (2.2) and (3.1), we have
p ( q p + 2 ) C 1 + p ( q p + 2 ) C 1 μ ϱ u n p ( q p + 2 ) φ ( u n ) p ( q p + 2 ) ϱ φ ( u n ) , u n = ( ϱ p ) ( q p + 2 ) ϱ u ˙ n p p + p ( q p + 2 ) R [ W 2 ( t , u n ( t ) ) 1 ϱ ( W 2 ( t , u n ( t ) ) , u n ( t ) ) ] d t p ( q p + 2 ) R [ W 1 ( t , u n ( t ) ) 1 ϱ ( W 1 ( t , u n ( t ) ) , u n ( t ) ) ] d t + p ( 1 q p + 2 ϱ ) R a ( t ) | u n ( t ) | q p + 2 d t ( ϱ p ) ( q p + 2 ) ϱ u ˙ n p p + p ( 1 q p + 2 ϱ ) u n q p + 2 , a q p + 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equal_HTML.gif
which implies that there exists a constant C 3 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq84_HTML.gif such that (3.3) holds. Next, we prove Step 2 still holds. From (W5)′, there exists δ ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq101_HTML.gif such that
| W ( t , x ) | 1 p a ( t ) | x | q p + 1 for  t R , | x | δ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ31_HTML.gif
(3.23)
By (3.23), we have
| W ( t , x ) | 1 p ( q p + 2 ) a ( t ) | x | q p + 2 for  t R , | x | δ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ32_HTML.gif
(3.24)
Let 0 < σ δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq130_HTML.gif and u = σ / C 2 : = ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq103_HTML.gif, it follows from (3.2) that
u C 2 u σ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equam_HTML.gif
which shows that | u ( t ) | σ δ < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq104_HTML.gif. It follows from (2.1) and (3.24) that
φ ( u ) = 1 p R | u ˙ ( t ) | p d t + R a ( t ) q p + 2 | u ( t ) | q p + 2 d t R W ( t , u ( t ) ) d t 1 p u ˙ p p + 1 q p + 2 u q p + 2 , a q p + 2 R 1 p ( q p + 2 ) a ( t ) | u ( t ) | q p + 2 d t = 1 p u ˙ p p + p 1 p ( q p + 2 ) u q p + 2 , a q p + 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equan_HTML.gif

Therefore, we can choose a constant α > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq105_HTML.gif depending on ρ such that φ ( u ) α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq106_HTML.gif for any u E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq107_HTML.gif with u = ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq108_HTML.gif. The proof of Theorem 1.2 is complete. □

Proof of Theorem 1.3 Condition (W4) shows that φ is even. In view of the proof of Theorem 1.1, we know that φ C 1 ( E , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq131_HTML.gif and satisfies (PS)-condition and assumptions (i) of Lemma 2.3. Now, we prove that (iii) of Lemma 2.4. Let E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq132_HTML.gif be a finite dimensional subspace of E. Since all norms of a finite dimensional space are equivalent, there exists c > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq133_HTML.gif such that
u c u . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ33_HTML.gif
(3.25)
Assume that dim E = m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq134_HTML.gif and { u 1 , u 2 , , u m } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq135_HTML.gif is a base of E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq132_HTML.gif such that
u i = c , i = 1 , 2 , , m . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ34_HTML.gif
(3.26)
For any u E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq136_HTML.gif, there exists λ i R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq137_HTML.gif, i = 1 , 2 , , m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq138_HTML.gif such that
u ( t ) = i = 1 m λ i u i ( t ) for  t R . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ35_HTML.gif
(3.27)
Let
u = i = 1 m | λ i | u i . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ36_HTML.gif
(3.28)
It is easy to see that https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq139_HTML.gif is a norm of E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq132_HTML.gif. Hence, there exists a constant c > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq140_HTML.gif such that c u u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq141_HTML.gif. Since u i E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq142_HTML.gif, by Lemma 2.2, we can choose R 1 > R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq143_HTML.gif such that
| u i ( t ) | < c δ 1 + c , | t | > R 1 , i = 1 , 2 , , m , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ37_HTML.gif
(3.29)
where δ is given in (3.24). Let
Θ = { i = 1 m λ i u i ( t ) : λ i R , i = 1 , 2 , , m ; i = 1 m | λ i | = 1 } = { u E : u = c } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ38_HTML.gif
(3.30)
Hence, for u Θ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq144_HTML.gif, let t 0 = t 0 ( u ) R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq145_HTML.gif such that
| u ( t 0 ) | = u . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ39_HTML.gif
(3.31)
Then by (3.25)-(3.28), (3.30) and (3.31), we have
c c = c c i = 1 m | λ i | = c i = 1 m | λ i | u i = c u u c u = c | u ( t 0 ) | c i = 1 m | λ i | | u i ( t 0 ) | , u Θ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ40_HTML.gif
(3.32)
This shows that | u ( t 0 ) | c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq146_HTML.gif and there exists i 0 { 1 , 2 , , m } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq147_HTML.gif such that | u i 0 ( t 0 ) | c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq148_HTML.gif, which together with (3.29), implies that | t 0 | R 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq149_HTML.gif. Let R 2 = R 1 + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq150_HTML.gif and
γ = min { W 1 ( t , x ) : R 2 t R 2 , c 2 1 / p | x | c C 2 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ41_HTML.gif
(3.33)
Since W 1 ( t , x ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq151_HTML.gif for all t R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq8_HTML.gif and x R N { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq152_HTML.gif, and W 1 C 1 ( R × R N , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq153_HTML.gif, it follows that γ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq154_HTML.gif. For any u E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq107_HTML.gif, from Lemma 2.5(i) and (3.2), we have
R 2 R 2 W 2 ( t , u ( t ) ) d t = { t [ R 2 , R 2 ] : | u ( t ) | > 1 } W 2 ( t , u ( t ) ) d t + { t [ R 2 , R 2 ] : | u ( t ) | 1 } W 2 ( t , u ( t ) ) d t { t [ R 2 , R 2 ] : | u ( t ) | > 1 } W 2 ( t , u ( t ) | u ( t ) | ) | u ( t ) | ϱ d t + R 2 R 2 max | x | 1 W 2 ( t , x ) d t u ϱ R 2 R 2 max | x | = 1 W 2 ( t , x ) d t + R 2 R 2 max | x | 1 W 2 ( t , x ) d t C 2 ϱ u ϱ R 2 R 2 max | x | = 1 W 2 ( t , x ) d t + R 2 R 2 max | x | 1 W 2 ( t , x ) d t = C 10 u ϱ + C 11 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ42_HTML.gif
(3.34)
where C 10 = C 2 ϱ R 2 R 2 max | x | = 1 W 2 ( t , x ) d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq155_HTML.gif, C 11 = R 2 R 2 max | x | 1 W 2 ( t , x ) d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq156_HTML.gif. Since u ˙ i L p ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq157_HTML.gif, i = 1 , 2 , , m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq138_HTML.gif, it follows that there exists ε ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq158_HTML.gif such that
t + ε t ε | u ˙ i ( s ) | d s ( 2 ε ) 1 / p ( t + ε t ε | u ˙ i ( s ) | p d s ) 1 / p ( 2 ε ) 1 / p u ˙ i p c 2 p for  t R , i = 1 , 2 , , m , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ43_HTML.gif
(3.35)
where 1 / p + 1 / p = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq159_HTML.gif. Then, for u Θ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq144_HTML.gif with | u ( t 0 ) | = u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq160_HTML.gif and t [ t 0 ε , t 0 + ε ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq161_HTML.gif, it follows from (3.27), (3.30), (3.31), (3.32) and (3.35) that
| u ( t ) | p = | u ( t 0 ) | p + p t 0 t | u ( s ) | p 2 ( u ˙ ( s ) , u ( s ) ) d s | u ( t 0 ) | p p t 0 ε t 0 + ε | u ( s ) | p 1 | u ˙ ( s ) | d s | u ( t 0 ) | p p | u ( t 0 ) | p 1 t 0 ε t 0 + ε | u ˙ ( s ) | d s c 2 | u ( t 0 ) | p 1 c p 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ44_HTML.gif
(3.36)
On the other hand, since u c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq162_HTML.gif for u Θ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq144_HTML.gif, then
| u ( t ) | u C 2 c , t R , u Θ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ45_HTML.gif
(3.37)
Therefore, from (3.33), (3.36) and (3.37), we have
R 2 R 2 W 1 ( t , u ( t ) ) d t t 0 ε t 0 + ε W 1 ( t , u ( t ) ) d t 2 ε γ for  u Θ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ46_HTML.gif
(3.38)
From (3.29) and (3.30), we have
| u ( t ) | i = 1 m | λ i | | u i ( t ) | δ for  | t | R 1 , u Θ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ47_HTML.gif
(3.39)
By (2.1), (3.15), (3.34), (3.38), (3.39) and Lemma 2.5, we have for u Θ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq144_HTML.gif and r > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq163_HTML.gif
φ ( r u ) = r p p u ˙ p p + r q p + 2 q p + 2 u q p + 2 , a q p + 2 + R [ W 2 ( t , r u ( t ) ) W 1 ( t , r u ( t ) ) ] d t r p p u ˙ p p + r q p + 2 q p + 2 u q p + 2 , a q p + 2 + r ϱ R W 2 ( t , u ( t ) ) d t r μ R W 1 ( t , u ( t ) ) d t = r p p u ˙ p p + r q p + 2 q p + 2 u q p + 2 , a q p + 2 + r ϱ R ( R 2 , R 2 ) W 2 ( t , u ( t ) ) d t r μ R ( R 2 , R 2 ) W 1 ( t , u ( t ) ) d t + r ϱ R 2 R 2 W 2 ( t , u ( t ) ) d t r μ R 2 R 2 W 1 ( t , u ( t ) ) d t r p p u ˙ p p + r q p + 2 q p + 2 u q p + 2 , a q p + 2 r ϱ R ( R 2 , R 2 ) W ( t , u ( t ) ) d t r μ R 2 R 2 W 1 ( t , u ( t ) ) d t + r ϱ R 2 R 2 W 2 ( t , u ( t ) ) d t r p p u ˙ p p + r q p + 2 q p + 2 u q p + 2 , a q p + 2 + r ϱ p ( q p + 2 ) R ( R 2 , R 2 ) a ( t ) | u ( t ) | q p + 2 d t + r ϱ ( C 10 u ϱ + C 11 ) 2 ε γ r μ r p p u ˙ p p + r q p + 2 q p + 2 u q p + 2 , a q p + 2 + r ϱ p ( q p + 2 ) u q p + 2 , a q p + 2 + r ϱ ( C 10 u ϱ + C 11 ) 2 ε γ r μ r p p c p + r q p + 2 q p + 2 c q p + 2 + r ϱ p ( q p + 2 ) c q p + 2 + C 10 ( r c ) ϱ + C 11 r ϱ 2 ε γ r μ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ48_HTML.gif
(3.40)
Since μ > ϱ > q p + 2 > p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq164_HTML.gif, we deduce that there exists r 0 = r 0 ( c , c , C 10 , C 11 , R 1 , R 2 , ε , γ ) = r 0 ( E ) > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq165_HTML.gif such that
φ ( r u ) < 0 for  u Θ  and  r r 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equao_HTML.gif
It follows that
φ ( u ) < 0 for  u E  and  u c r 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equap_HTML.gif
which shows that (iii) of Lemma 2.4 holds. By Lemma 2.4, φ possesses an unbounded sequence { d n } n = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq166_HTML.gif of critical values with d n = φ ( u n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq167_HTML.gif, where u n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq168_HTML.gif is such that φ ( u n ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq169_HTML.gif for n = 1 , 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq170_HTML.gif . If { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq171_HTML.gif is bounded, then there exists C 12 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq172_HTML.gif such that
u n C 12 for  n N . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ49_HTML.gif
(3.41)
By (3.2) and (3.41), we get
| u n ( t ) | C 2 C 12 for  n N . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ50_HTML.gif
(3.42)
From (W5), we can choose C 13 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq173_HTML.gif and R 3 > R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq174_HTML.gif such that
| W ( t , x ) | C 13 a ( t ) | x | q p + 1 for  | t | R 3 , | x | C 2 C 12 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equaq_HTML.gif
which implies that
| W ( t , x ) | C 13 q p + 2 a ( t ) | x | q p + 2 for  | t | R 3 , | x | C 2 C 12 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ51_HTML.gif
(3.43)
Hence, by (2.1) and (3.43), we have
1 p u ˙ n p p + 1 q p + 2 u n q p + 2 , a q p + 2 = d n + R W ( t , u n ( t ) ) d t = d n + R [ R 3 , R 3 ] W ( t , u n ( t ) ) d t + R 3 R 3 W ( t , u n ( t ) ) d t d n C 13 q p + 2 R [ R 3 , R 3 ] a ( t ) | u n ( t ) | q p + 2 d t R 3 R 3 | W ( t , u n ( t ) ) | d t d n C 13 q p + 2 u n q p + 2 , a q p + 2 R 3 R 3 max | x | C 2 C 12 | W ( t , x ) | d t , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equar_HTML.gif
which, together with (3.41), implies that
d n 1 p u ˙ n p p + C 13 + 1 q p + 2 u n q p + 2 , a q p + 2 + R 3 R 3 max | x | C 2 C 12 | W ( t , x ) | d t < + . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equas_HTML.gif

This contradicts the fact that { d n } n = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq166_HTML.gif is unbounded, and so { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq171_HTML.gif is unbounded. The proof is complete. □

Proof of Theorem 1.4 In view of the proofs of Theorem 1.2 and Theorem 1.3, the conclusion of Theorem 1.4 holds. The proof is complete. □

4 Examples

Example 4.1 Consider the following system:
d d t ( | u ˙ ( t ) | u ˙ ( t ) ) a ( t ) | u ( t ) | 3 u ( t ) + W ( t , u ( t ) ) = 0 , a.e.  t R , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ52_HTML.gif
(4.1)
where p = 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq175_HTML.gif, q = 6 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq176_HTML.gif, t R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq8_HTML.gif, u R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq9_HTML.gif, a C ( R , ( 0 , ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq177_HTML.gif and a satisfies (A)′. Let
W ( t , x ) = a ( t ) ( i = 1 m a i | x | μ i j = 1 n b j | x | ϱ j ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equat_HTML.gif
where μ 1 > μ 2 > > μ m > ϱ 1 > ϱ 2 > > ϱ j > 5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq178_HTML.gif, a i , b j > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq179_HTML.gif, i = 1 , , m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq180_HTML.gif, j = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq181_HTML.gif. Let
W 1 ( t , x ) = a ( t ) i = 1 m a i | x | μ i , W 2 ( t , x ) = a ( t ) j = 1 n b j | x | ϱ j . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equau_HTML.gif

Then it is easy to check that all the conditions of Theorem 1.3 are satisfied with μ = μ m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq182_HTML.gif and ϱ = ϱ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq183_HTML.gif. Hence, problem (4.1) has an unbounded sequence of homoclinic solutions.

Example 4.2 Consider the following system:
d d t ( | u ˙ ( t ) | 1 / 2 u ˙ ( t ) ) a ( t ) | u ( t ) | u ( t ) + W ( t , u ( t ) ) = 0 , a.e.  t R , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equ53_HTML.gif
(4.2)
where p = 3 / 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq184_HTML.gif, q = 5 / 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq185_HTML.gif, t R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq8_HTML.gif, u R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq9_HTML.gif, a C ( R , ( 0 , ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq177_HTML.gif and a satisfies (A)′. Let
W ( t , x ) = a ( t ) [ a 1 | x | μ 1 + a 2 | x | μ 2 b 1 ( cos t ) | x | ϱ 1 b 2 | x | ϱ 2 ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equav_HTML.gif
where μ 1 > μ 2 > ϱ 1 > ϱ 2 > 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq186_HTML.gif, a 1 , a 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq187_HTML.gif, b 1 , b 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq188_HTML.gif. Let
W 1 ( t , x ) = a ( t ) ( a 1 | x | μ 1 + a 2 | x | μ 2 ) , W 2 ( t , x ) = a ( t ) [ b 1 ( cos t ) | x | ϱ 1 + b 2 | x | ϱ 2 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_Equaw_HTML.gif

Then it is easy to check that all the conditions of Theorem 1.4 are satisfied with μ = μ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq189_HTML.gif and ϱ = ϱ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-137/MediaObjects/13661_2012_Article_390_IEq183_HTML.gif. Hence, by Theorem 1.4, problem (4.2) has an unbounded sequence of homoclinic solutions.

Declarations

Acknowledgements

XS and QZ are supported by the Scientific Research Foundation of Guangxi Education Office (No. 201203YB093), Guangxi Natural Science Foundation (Nos. 2013GXNSFBA019004 and 2012GXNSFBA053013) and the Scientific Research Foundation of Guilin University of Technology. QMZ is supported by the NNSF of China (No. 11201138) and the Scientific Research Fund of Hunan Provincial Education Department (No. 12B034).

Authors’ Affiliations

(1)
College of Science, Guilin University of Technology
(2)
College of Science, Hunan University of Technology

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