Open Access

On symmetric positive homoclinic solutions of semilinear p-Laplacian differential equations

Boundary Value Problems20122012:121

DOI: 10.1186/1687-2770-2012-121

Received: 31 July 2012

Accepted: 8 October 2012

Published: 24 October 2012

Abstract

In this paper we study the existence of even positive homoclinic solutions for p-Laplacian ordinary differential equations (ODEs) of the type ( u | u | p 2 ) a ( x ) u | u | p 2 + λ b ( x ) u | u | q 2 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq1_HTML.gif, where 2 p < q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq2_HTML.gif, λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq3_HTML.gif and the functions a and b are strictly positive and even. First, we prove a result on symmetry of positive solutions of p-Laplacian ODEs. Then, using the mountain-pass theorem, we prove the existence of symmetric positive homoclinic solutions of the considered equations. Some examples and additional comments are given.

MSC:34B18, 34B40, 49J40.

Keywords

p-Laplacian ODEs homoclinic solution weak solution Palais-Smale condition mountain-pass theorem

1 Introduction and main results

In this paper we prove the existence of positive homoclinic solutions for p-Laplacian ODEs of the type
( u | u | p 2 ) a ( x ) u | u | p 2 + λ b ( x ) u | u | q 2 = 0 , x R , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equ1_HTML.gif
(1)

where 2 p < q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq2_HTML.gif and λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq3_HTML.gif. We assume that

(H) the functions a ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq4_HTML.gif are b ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq5_HTML.gif are continuously differentiable, strictly positive, 0 < a a ( x ) A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq6_HTML.gif and 0 < b b ( x ) B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq7_HTML.gif. Let, moreover, a ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq4_HTML.gif and b ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq5_HTML.gif be even functions on , x a ( x ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq8_HTML.gif and x b ( x ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq9_HTML.gif for x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq10_HTML.gif.

By a solution of (1), we mean a function u : R R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq11_HTML.gif such that u C 1 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq12_HTML.gif, ( u | u | p 2 ) C ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq13_HTML.gif and Eq. (1) holds for every x R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq14_HTML.gif. We are looking for positive solutions of (1) which are homoclinic, i.e., u ( x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq15_HTML.gif and u ( x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq16_HTML.gif as | x | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq17_HTML.gif.

In the case p = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq18_HTML.gif, q = 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq19_HTML.gif and λ = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq20_HTML.gif, similar problems are considered in [13] using variational methods. Note that in [2] and [3] the following second-order differential equations are considered:
u a ( x ) u b ( x ) u 2 + c ( x ) u 3 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equa_HTML.gif
and
u + a ( x ) u b ( x ) u 2 + c ( x ) u 3 = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equb_HTML.gif

where a, b and c are periodic, bounded functions and a and c are positive. These equations come from a biomathematics model suggested by Austin [4] and Cronin [5]. Further results and the phase plane analysis of these equations with constant coefficients are given in [6]. Note that the periodic and homoclinic solutions of p-Laplacian ODEs are considered in [7, 8].

The present work is an extension of these studies to p-Laplacian ODEs. Let X T : = W 0 1 , p ( T , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq21_HTML.gif be the Sobolev space of p-integrable absolutely continuous functions u : [ T , T ] R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq22_HTML.gif such that
u p = T T ( | u ( x ) | p + | u ( x ) | p ) d x < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equc_HTML.gif

and u ( T ) = u ( T ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq23_HTML.gif.

We use a variational treatment of the problem considering the functional J T : X T R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq24_HTML.gif
J T ( u ) = T T ( 1 p ( | u ( x ) | p + a ( x ) | u ( x ) | p ) λ q b ( x ) ( u + ( x ) ) q ) d x , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equd_HTML.gif

where u + ( x ) = max { u ( x ) , 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq25_HTML.gif.

Using the well-known mountain-pass theorem, we conclude that the functional J T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq26_HTML.gif has a nontrivial critical point u T , λ X T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq27_HTML.gif, which is a solution of the restricted problem
( u | u | p 2 ) a ( x ) u | u | p 2 + λ b ( x ) u | u | q 2 = 0 , x ( T , T ) , u ( T ) = u ( T ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equ2_HTML.gif
(2)

Further, we obtain uniform estimates for the solutions u T , λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq28_HTML.gif, extended by 0 outside [ T , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq29_HTML.gif. Then, a positive homoclinic solution u λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq30_HTML.gif of (1) is found as a limit of u T , λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq28_HTML.gif, as T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq31_HTML.gif in C loc 1 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq32_HTML.gif. The function u λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq30_HTML.gif is also an even function.

To obtain the property, we extend the symmetry lemma of Korman and Ouyang [9] to the p-Laplacian equations. The result is formulated and proved in Section 2.

Our main result is:

Theorem 1 Suppose that 2 p < q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq2_HTML.gif, λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq3_HTML.gif and assumptions (H) hold. Then Eq. (1) has a positive solution u λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq30_HTML.gif such that u λ ( x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq33_HTML.gif and u λ ( x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq34_HTML.gif as | x | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq35_HTML.gif. Moreover, the solution u λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq30_HTML.gif is an even function, max { u λ ( x ) : x R } = u λ ( 0 ) + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq36_HTML.gif as λ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq37_HTML.gif and u λ ( x ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq38_HTML.gif for x > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq39_HTML.gif.

Theorem 1 is proved in Section 3. From its proof we have
max { u λ ( x ) : x R } = u λ ( 0 ) ( a ( 0 ) λ b ( 0 ) ) 1 / ( q p ) > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Eque_HTML.gif
from which it follows that u λ ( 0 ) + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq40_HTML.gif as λ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq37_HTML.gif. Observe that if λ = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq41_HTML.gif, the problem
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equf_HTML.gif
has a unique solution u = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq42_HTML.gif. Indeed, multiplying the equation by u and integrating by parts over , we obtain
( | u ( x ) | p + a ( x ) | u ( x ) | p ) d x = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equg_HTML.gif

which implies that u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq43_HTML.gif.

A simplified method can be applied to the equations
u a ( x ) u | u | p 2 + λ b ( x ) u | u | q 2 = 0 , x R , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equ3_HTML.gif
(3)
under assumptions (H) and 2 p < q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq2_HTML.gif, λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq3_HTML.gif. Note that in this case, the even homoclinic solution u λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq30_HTML.gif of Eq. (3) satisfies
max { u λ ( x ) : x R } = u λ ( 0 ) ( a ( 0 ) λ b ( 0 ) ) 1 / ( q p ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equh_HTML.gif

and again u λ ( 0 ) + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq40_HTML.gif as λ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq37_HTML.gif. If a and b are constants, Eq. (3) is a conservative system and one can plot the phase curves ( v 2 ) 2 a | u | p p + λ b | u | q q = C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq44_HTML.gif in the phase plane ( u , v ) = ( u , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq45_HTML.gif. An example is given at the end of Section 3.

2 Preliminary results

Let φ p ( t ) = t | t | p 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq46_HTML.gif, p 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq47_HTML.gif and Φ p ( t ) = | t | p p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq48_HTML.gif. It is clear that Φ p ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq49_HTML.gif is a differentiable function and Φ p ( t ) = φ p ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq50_HTML.gif. Moreover, φ p ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq51_HTML.gif exists and φ p ( t ) = ( p 1 ) | t | p 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq52_HTML.gif for p 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq47_HTML.gif.

Let L p ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq53_HTML.gif, 1 < p < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq54_HTML.gif be the space of Lebesgue measurable functions u : ( a , b ) R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq55_HTML.gif such that the norm | u | p p = a b | u ( x ) | p d x < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq56_HTML.gif.

The dual space of L p ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq53_HTML.gif is L p ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq57_HTML.gif, where 1 p + 1 p = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq58_HTML.gif. Let , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq59_HTML.gif be the duality pairing between L p ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq57_HTML.gif and L p ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq53_HTML.gif. By the Hölder inequality, | v , u | | v | p | u | p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq60_HTML.gif for any v L p ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq61_HTML.gif and u L p ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq62_HTML.gif. We will use the following lemmata in further considerations.

Lemma 2 For any u , v L p ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq63_HTML.gif, the following inequality holds:
φ p ( u ) φ p ( v ) , u v ( | u | p p 1 | v | p p 1 ) ( | u | p | v | p ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equi_HTML.gif
Proof of Lemma 2. Note that for u L p ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq62_HTML.gif, φ p ( u ) L p ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq64_HTML.gif. From the Hölder inequality, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equj_HTML.gif

 □

Lemma 3 Let p 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq47_HTML.gif, u C 1 ( [ a , b ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq65_HTML.gif and ( u | u | p 2 ) C ( [ a , b ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq66_HTML.gif. Then
a b ( u | u | p 2 ) u d x = p 1 p ( | u ( b ) | p | u ( a ) | p ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equk_HTML.gif
The statement of Lemma 3 follows simply from the identity
( | u | p ) = p p 1 ( u | u | p 2 ) u . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equl_HTML.gif
The one-dimensional p-Laplacian operator L p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq67_HTML.gif for a differentiable function u on the interval I is introduced as L p ( u ) : = ( φ p ( u ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq68_HTML.gif. Let us consider the problem
{ L p ( u ) + f ( x , u ) = 0 , x ( T , T ) , u ( T ) = u ( T ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equ4_HTML.gif
(4)
where f C 1 ( [ T , T ] × R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq69_HTML.gif and satisfies
f ( x , u ) = f ( x , u ) , x ( T , T ) , u > 0 , x f x ( x , u ) < 0 , x ( T , T ) { 0 } , u > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equ5_HTML.gif
(5)

A function u : [ T , T ] R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq22_HTML.gif is said to be a solution of the problem (4) if u C 1 ( [ T , T ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq70_HTML.gif with u ( T ) = u ( T ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq23_HTML.gif is such that u | u | p 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq71_HTML.gif is absolutely continuous and L p u ( x ) + f ( x , u ( x ) ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq72_HTML.gif holds a.e. in ( T , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq73_HTML.gif.

We formulate an extension of Lemma 1 of [9] for p-Laplacian nonlinear equations. The result of Korman and Ouyang is one-dimensional analogue of the result of Gidas, Ni and Nirenberg [10] for symmetry of positive solutions of semilinear Laplace equations. In the case of p-Laplacian equations, the symmetry of solutions in higher dimensions is discussed by Reihel and Walter [11].

Theorem 4 Assume that f C 1 ( [ T , T ] × R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq74_HTML.gif satisfies (5). Then any positive solution u of (4) is an even function such that max { u ( x ) : T x T } = u ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq75_HTML.gif and u ( x ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq76_HTML.gif for x ( 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq77_HTML.gif.

Remark 1 Let us note that if the function f satisfies (5), but u is not a positive solution of (4), then u is not necessarily an even function. A simple counter example in the case p = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq18_HTML.gif is the problem
{ u + u x 2 + π 2 2 = 0 , π < x < π , u ( π ) = u ( π ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equm_HTML.gif
The term f ( x , u ) = u x 2 + π 2 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq78_HTML.gif satisfies (5) in the interval ( π , π ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq79_HTML.gif, but the solution of the problem u ( x ) = x 2 π 2 + sin x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq80_HTML.gif is negative in ( π , π ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq79_HTML.gif and not an even function. Its graph is presented in Figure 1. It would be more interesting to show an example for the case p > 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq81_HTML.gif and f satisfying the additional assumption f ( x , 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq82_HTML.gif.
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Fig1_HTML.jpg
Figure 1

Graph of the functions u ( x ) = x 2 π 2 + sin x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq83_HTML.gif .

Sketch of Proof of Theorem 4 Suppose that the function u has only one global maximum on [ T , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq84_HTML.gif.

Assume that the function u ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq85_HTML.gif has a finite number of local minima in the interval [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq86_HTML.gif, and let x 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq87_HTML.gif be the largest local minimum. Let x ¯ [ x 1 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq88_HTML.gif be the local maximum and x ˜ [ x ¯ , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq89_HTML.gif be such that u ( x 1 ) = u ( x ˜ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq90_HTML.gif. Denote u 1 = u ( x 1 ) = u ( x ˜ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq91_HTML.gif and u 2 = u ( x ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq92_HTML.gif, and let x = α ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq93_HTML.gif and x = β ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq94_HTML.gif be the inverse functions of the function u = u ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq95_HTML.gif in the intervals [ x 1 , x ¯ ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq96_HTML.gif and [ x ¯ , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq97_HTML.gif, respectively. Multiplying the equation in (4) by u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq98_HTML.gif and integrating in [ x 1 , x ˜ ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq99_HTML.gif, we obtain by Lemma 3 and (5):
0 = x 1 x ˜ ( L p ( u ) u + f ( x , u ) u ) d x = p 1 p | u | p ( x ˜ ) + x 1 x ¯ f ( x , u ) u d x + x ¯ x ˜ f ( x , u ) u d x = p 1 p | u | p ( x ˜ ) + u 1 u 2 ( f ( α ( u ) , u ) f ( β ( u ) , u ) ) d u > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equn_HTML.gif

which leads to contradiction. One can prove the last fact using other arguments; see, for instance, Theorem 2.1 of [12]. Suppose now that u has infinitely many local minima in [ T , x ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq100_HTML.gif. Further, we can follow the steps of the proof of Lemma 1 of [9] with corresponding modifications based on Lemma 3. □

3 Proof of the main result

Let X T = W 0 1 , p ( T , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq101_HTML.gif be the Sobolev space of p-integrable absolutely continuous functions u : [ T , T ] R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq22_HTML.gif such that
u T p = T T ( | u ( x ) | p + | u ( x ) | p ) d x < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equo_HTML.gif

and u ( T ) = u ( T ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq23_HTML.gif. Note that if a ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq4_HTML.gif is strictly positive and bounded, i.e., there exist a and A such that 0 < a a ( x ) A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq6_HTML.gif, then u a , T p = T T ( | u ( x ) | p + a ( x ) | u ( x ) | p ) d x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq102_HTML.gif is an equivalent norm in X T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq103_HTML.gif.

We need an extension to the p-case of the following proposition by Rabinowitz [13].

Proposition 5 Let u W loc 1 , p ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq104_HTML.gif. Then:
  1. (i)
    If T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq105_HTML.gif, for x [ T 1 / 2 , T + 1 / 2 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq106_HTML.gif,
    max x [ T 1 / 2 , T + 1 / 2 ] | u ( x ) | 2 p 1 p ( T 1 / 2 T + 1 / 2 ( | u ( t ) | p + | u ( t ) | p ) d t ) 1 / p . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equ6_HTML.gif
    (6)
     
  2. (ii)
    For every u W 0 1 , p ( T , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq107_HTML.gif,
    u L ( T , T ) 2 p 1 p u T . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equ7_HTML.gif
    (7)
     
Proof of Proposition 5 Let x , t [ T 1 / 2 , T + 1 / 2 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq108_HTML.gif. It follows
| u ( x ) | | u ( t ) | + T 1 / 2 T + 1 / 2 | u ( s ) | d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equp_HTML.gif
Integrating with respect to t [ T 1 / 2 , T + 1 / 2 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq109_HTML.gif and using the Hölder and Jensen inequalities, we obtain
| u ( x ) | T 1 / 2 T + 1 / 2 | u ( t ) | d t + T 1 / 2 T + 1 / 2 | u ( s ) | d s ( T 1 / 2 T + 1 / 2 | u ( t ) | p d t ) 1 / p + ( T 1 / 2 T + 1 / 2 | u ( t ) | p d t ) 1 / p 2 p 1 p ( T 1 / 2 T + 1 / 2 ( | u ( t ) | p + | u ( t ) | p ) d t ) 1 / p . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equq_HTML.gif
  1. (ii)
    Take u W 0 1 , p ( T , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq110_HTML.gif. Since W 0 1 , p ( T , T ) C [ T , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq111_HTML.gif, there exists τ [ T , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq112_HTML.gif such that by (i)
    u L ( T , T ) = u C [ τ 1 / 2 , τ + 1 / 2 ] 2 p 1 p ( τ 1 / 2 τ + 1 / 2 ( | u ( t ) | p + | u ( t ) | p ) d t ) 1 / p 2 u . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equr_HTML.gif
     

 □

We are looking for positive solutions of (1), which are homoclinic, i.e., u ( x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq15_HTML.gif and u ( x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq16_HTML.gif as | x | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq35_HTML.gif. Firstly, we look for positive solutions of the problem
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equs_HTML.gif

A function u : [ T , T ] R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq22_HTML.gif is said to be a solution of the problem ( P T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq113_HTML.gif) if u C 1 ( [ T , T ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq70_HTML.gif with u ( T ) = u ( T ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq23_HTML.gif is such that φ p ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq114_HTML.gif is absolutely continuous and ( φ p ( u ) ) ( x ) a ( x ) φ p ( u ) ( x ) + λ b ( x ) φ q ( u ) ( x ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq115_HTML.gif holds a.e. in ( T , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq73_HTML.gif.

A function u : [ T , T ] R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq22_HTML.gif is said to be a weak solution of the problem ( P T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq113_HTML.gif) if
T T ( ( φ p ( u ) ) v d x + a ( x ) φ p ( u ) v λ b ( x ) φ q ( u ) v ) d x = 0 , v W 0 1 , p ( ( T , T ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equt_HTML.gif
Standard arguments show that a weak solution of the problem ( P T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq113_HTML.gif) is a solution of ( P T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq113_HTML.gif) (see [14] and [15]). Consider the modified problem
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equu_HTML.gif
where u + = max ( u , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq116_HTML.gif. It is easy to see that solutions of the problem ( P T + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq117_HTML.gif) are positive solutions of the problem ( P T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq113_HTML.gif). Indeed, if u ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq85_HTML.gif is a solution of ( P T + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq117_HTML.gif) and u ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq85_HTML.gif has negative minimum at x 0 ( T , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq118_HTML.gif, since for p 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq47_HTML.gif, ( φ p ( u ) ) ( x 0 ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq119_HTML.gif, by the equation ( φ p ( u ) ) a ( x ) φ p ( u ) + λ b ( x ) ( u + ) q 1 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq120_HTML.gif, we reach a contradiction
0 = ( φ p ( u ) ) ( x 0 ) + a ( x 0 ) ( u ( x 0 ) ) p 1 > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equv_HTML.gif
Then u ( x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq121_HTML.gif and u is a solution of ( P T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq113_HTML.gif). We use a variational treatment of the problem ( P T + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq117_HTML.gif), considering the functional J T : X T R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq24_HTML.gif
J T ( u ) = T T ( 1 p ( | u ( x ) | p + a ( x ) | u ( x ) | p ) λ q b ( x ) ( u + ( x ) ) q ) d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equw_HTML.gif
Critical points of J T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq122_HTML.gif are weak solutions of ( P T + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq117_HTML.gif), i.e.,
T T ( φ p ( u ) v + a ( x ) φ p ( u ) v λ b ( x ) ( u + ) q 1 v ) d x , v W 0 1 , p ( T , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equx_HTML.gif

and, by a standard way, they are solutions of ( P T + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq117_HTML.gif). We show that J T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq122_HTML.gif satisfies the assumptions of the mountain-pass theorem of Ambrosetti and Rabinowitz [16].

Theorem 6 (Mountain-pass theorem)

Let X be a Banach space with norm https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq123_HTML.gif, I C 1 ( X , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq124_HTML.gif, I ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq125_HTML.gif and I satisfy the (PS) condition. Suppose that there exist r > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq126_HTML.gif, α > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq127_HTML.gif and e X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq128_HTML.gif such that e > r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq129_HTML.gif
  1. (i)

    I ( x ) α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq130_HTML.gif if x = r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq131_HTML.gif,

     
  2. (ii)
    I ( e ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq132_HTML.gif. Let c = inf γ Γ { max 0 t 1 I ( γ ( t ) ) } α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq133_HTML.gif, where
    Γ = { γ C ( [ 0 , 1 ] , X ) : γ ( 0 ) = 0 , γ ( 1 ) = e } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equy_HTML.gif
     

Then c is a critical value of I, i.e., there exists x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq134_HTML.gif such that I ( x 0 ) = c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq135_HTML.gif and I ( x 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq136_HTML.gif.

Next, denote by C j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq137_HTML.gif several positive constants.

Lemma 7 Let 2 p < q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq2_HTML.gif, λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq3_HTML.gif and assumptions (H) hold. Then for every T > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq138_HTML.gif, the problem ( P T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq113_HTML.gif) has a positive solution u T , λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq28_HTML.gif. Moreover, there is a constant K > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq139_HTML.gif, independent of T, such that
u T , λ T K . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equ8_HTML.gif
(8)

Proof Step 1. J T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq122_HTML.gif satisfies the (PS) condition.

Let ( u k ) k X T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq140_HTML.gif be a sequence, and suppose there exist C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq141_HTML.gif and k 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq142_HTML.gif such that for k k 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq143_HTML.gif
| J T ( u k ) | = | T T ( 1 p ( | u k ( x ) | p + a ( x ) | u k ( x ) | p ) λ q b ( x ) ( u k + ( x ) ) q ) d x | C 1 p , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equ9_HTML.gif
(9)
and
| J T ( u k ) , u k | = | T T ( | u k ( x ) | p + a ( x ) | u k ( x ) | p λ b ( x ) ( u k + ( x ) ) q ) d x | u k T . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equ10_HTML.gif
(10)
Let us denote a ˆ = min ( 1 , a ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq144_HTML.gif. From (9) and (10), it follows that
C 1 T T ( ( | u k ( x ) | p + a ( x ) | u k ( x ) | p ) λ p q b ( x ) ( u k + ( x ) ) q ) d x C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equz_HTML.gif
and
u k T T T ( | u k ( x ) | p a ( x ) | u k ( x ) | p + λ b ( x ) ( u k + ( x ) ) q ) d x u k T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equaa_HTML.gif
Then
C 1 + u k T λ ( q p ) b p T T ( u k + ( x ) ) q d x , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equab_HTML.gif
and
a ˆ u k T p C 1 T T ( | u k ( x ) | p + a ( x ) | u k ( x ) | p ) d x C 1 λ p q T T b ( x ) ( u k + ( x ) ) q d x λ p B q T T ( u k + ( x ) ) q d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equac_HTML.gif
We have
a ˆ u k T p C 1 B q ( q p ) b ( C 1 + u k T ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equad_HTML.gif
which implies that the sequence ( u k ) k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq145_HTML.gif is bounded in X T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq103_HTML.gif. By the compact embedding X T C ( [ T , T ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq146_HTML.gif, there exist u X T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq147_HTML.gif and the subsequence of ( u k ) k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq145_HTML.gif, still denoted by ( u k ) k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq145_HTML.gif, such that u k u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq148_HTML.gif weakly in X T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq103_HTML.gif and u k u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq149_HTML.gif strongly in C ( [ T , T ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq150_HTML.gif. We will show that u k u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq149_HTML.gif strongly in X T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq103_HTML.gif using Lemma 2. By uniform convergence of u k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq151_HTML.gif to u in C ( [ T , T ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq152_HTML.gif, it follows that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equae_HTML.gif
and
φ p ( u k ) φ p ( u ) , a ( x ) ( u k u ) φ q ( u k ) φ q ( u ) , b ( x ) ( u k u ) 0 , k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equaf_HTML.gif
Then
φ p ( u k ) φ p ( u ) , u k u 0 , k , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equag_HTML.gif
and by Lemma 2,
φ p ( u k ) φ p ( u ) , u k u ( | u k | p p 1 | u | p p 1 ) ( | u k | p | u | p ) 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equah_HTML.gif

which implies that | u k | p | u | p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq153_HTML.gif. Then u k T u T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq154_HTML.gif and by the uniform convexity of the space X T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq103_HTML.gif, it follows that u k u T 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq155_HTML.gif, as k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq156_HTML.gif.

Step 2. Geometric conditions.

Obviously, J T ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq157_HTML.gif. By assumption (H) it follows
J T ( u ) a ˆ 2 p u T p + T T ( a ( x ) 2 p | u ( x ) | p λ p q b ( x ) ( u + ( x ) ) q ) d x a ˆ 2 p u T p + T T | u ( x ) | p ( a 2 p λ p q b ( x ) | u ( x ) | q p ) d x > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equai_HTML.gif

if u T = ρ : = ( a q 2 λ p 2 ) 1 / ( q p ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq158_HTML.gif. Then J T ( u ) a ˆ ρ p 2 p > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq159_HTML.gif.

Let u 0 ( x ) X T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq160_HTML.gif be such that u 0 ( x ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq161_HTML.gif if x ( T , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq162_HTML.gif and also u 0 ( T ) = u 0 ( T ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq163_HTML.gif. Consider the function
u ˆ 0 ( x ) = { μ u 0 ( x ) , if  x [ 1 , 1 ] , 0 , if  x [ T , T ] [ 1 , 1 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equaj_HTML.gif
Then
J T ( u ˆ 0 ) = μ p T T 1 p ( | u 0 ( x ) | p + a ( x ) | u 0 ( x ) | p ) d x μ q T T λ q b ( x ) ( u 0 ( x ) ) q d x < 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equak_HTML.gif

for μ large enough.

By the mountain-pass theorem, there exists a solution u T , λ X T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq164_HTML.gif such that
c T = J T ( u T , λ ) = inf γ Γ T max t [ 0 , 1 ] J T ( γ ( t ) ) , J T ( u T , λ ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equ11_HTML.gif
(11)
where
Γ T = { γ ( t ) C ( [ 0 , 1 ] , X T ) : γ ( 0 ) = 0 , γ ( 1 ) = u ˆ 0 ( x ) } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equal_HTML.gif
Moreover, using the variational characterization (11), we have
c T a ˆ ρ p 2 p > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equam_HTML.gif

Therefore, u T , λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq165_HTML.gif is a nontrivial and positive solution of ( P T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq113_HTML.gif). By Theorem 4, max { u T , λ : T x T } = u T , λ ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq166_HTML.gif and u T , λ ( x ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq167_HTML.gif for x ( 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq77_HTML.gif.

Step 3. Uniform estimates.

Let T 1 T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq168_HTML.gif. By continuation with zero of a function u X T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq147_HTML.gif to [ T 1 , T 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq169_HTML.gif, we have X T X T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq170_HTML.gifand Γ T Γ T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq171_HTML.gif. Using the variational characterization (11), we infer that c T 1 c T c 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq172_HTML.gif and then
T T ( 1 p ( | u T , λ ( x ) | p + a ( x ) u T , λ p ( x ) ) λ q b ( x ) u T , λ q ( x ) ) d x c 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equ12_HTML.gif
(12)
Multiplying the equation of ( P T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq113_HTML.gif) by u T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq173_HTML.gif and integrating by parts, we have
T T ( | u T , λ | p + a ( x ) u T , λ p ) d x = T T λ b ( x ) u T , λ q d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equan_HTML.gif
Then by (12),
c 1 T T ( 1 p ( | u T , λ | p + a ( x ) u T , λ p ) λ q λ b ( x ) u T , λ q ) d x ( 1 p 1 q ) T T ( | u T , λ | p + a ( x ) u T , λ p ) d x a ˆ ( q p ) p q u T , λ T p . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equao_HTML.gif

We get (8) with K = p q c 1 a ˆ ( q p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq174_HTML.gif, which completes the proof. □

Proof of Theorem 1 Take T n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq175_HTML.gif and let u n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq176_HTML.gif be the solution of the problem ( P T n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq177_HTML.gif) given by Lemma 2. Consider the extension of u n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq176_HTML.gif to with zero outside [ T n , T n ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq178_HTML.gif and denote it by the same symbol.

Claim 1. The sequence of functions ( u n ) n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq179_HTML.gif is uniformly bounded and equicontinuous.

By (8) and the embedding of X T n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq180_HTML.gif in C ( [ T n , T n ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq181_HTML.gif, there is K 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq182_HTML.gif such that u n L ( [ T n , T n ] ) K 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq183_HTML.gif. Then by the equation of ( P T n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq177_HTML.gif), it follows that
( φ p ( u n ) ) L ( [ T n , T n ] ) K 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equ13_HTML.gif
(13)
By the mean value theorem for every natural n and every t R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq184_HTML.gif, there exists ξ n [ t 1 , t ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq185_HTML.gif such that
u n ( t ) u n ( t 1 ) = u n ( ξ k ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equap_HTML.gif
Then, as a consequence of (13), we obtain
| φ p ( u n ( t ) ) | = | ξ k t ( φ p ( u n ( s ) ) ) d s + φ p ( u n ( ξ k ) ) | t 1 t | ( φ p ( u n ( s ) ) ) | d s + | u n ( ξ k ) | p 1 K 2 + ( | u n ( t ) | + | u n ( t 1 ) | ) p 1 K 2 + ( 2 K 1 ) p 1 = : K 3 ( p 1 ) / p , t R , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equ14_HTML.gif
(14)

from which it follows u n L ( [ T n , T n ] ) K 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq186_HTML.gif and the sequence of functions ( u n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq187_HTML.gif is equicontinuous. Further, we claim that the sequence ( u n ) n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq188_HTML.gif is also equicontinuous.

Claim 2. The sequence of functions ( u n ) n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq188_HTML.gif is equicontinuous.

To prove this statement, we follow the method given by Tang and Xiao [7]. For completeness, we present it in details.

Suppose that ( u n ) n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq188_HTML.gif is not an equicontinuous sequence in C loc ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq189_HTML.gif. Then there exist an ε 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq190_HTML.gif and sequences ( t k 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq191_HTML.gif and ( t k 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq192_HTML.gif such that 0 < t k 1 t k 2 < 1 k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq193_HTML.gif and
| u n ( t k 1 ) u n ( t k 2 ) | ε 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equ15_HTML.gif
(15)
By (14), there are numbers w 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq194_HTML.gif and w 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq195_HTML.gif and the subsequence ( u n k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq196_HTML.gif such that u n k ( t k 1 ) w 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq197_HTML.gif and u n k ( t k 2 ) w 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq198_HTML.gif as k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq156_HTML.gif. By (15), | w 1 w 2 | ε 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq199_HTML.gif. On the other hand, by (13) we have
| φ p ( u n k ( t k 2 ) ) φ p ( u n k ( t k 1 ) ) | t k 1 t k 2 | φ p ( u n k ( s ) ) | d s K 2 k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equaq_HTML.gif

Then passing to a limit as k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq156_HTML.gif, we obtain φ p ( w 1 ) = φ p ( w 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq200_HTML.gif. Hence, w 1 = w 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq201_HTML.gif which contradicts | w 1 w 2 | ε 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq199_HTML.gif. Thus, the sequence ( u n ) n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq188_HTML.gif is equicontinuous.

Let T > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq138_HTML.gif. By Claim 1 and Claim 2 and the Arzelà-Ascoli theorem, there is a subsequence of ( u n ) n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq179_HTML.gif, still denoted by ( u n ) n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq202_HTML.gif, and functions u λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq203_HTML.gif and v λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq204_HTML.gif of C ( [ T , T ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq205_HTML.gif such that u n u λ 1 C ( [ T , T ] ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq206_HTML.gif and u n v λ 1 C ( [ T , T ] ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq207_HTML.gif. Trivially, it follows that u λ 1 C 1 ( [ T , T ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq208_HTML.gif, u λ 1 = v λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq209_HTML.gif and u n v λ 1 C 1 ( [ T , T ] ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq210_HTML.gif. Repeating this procedure as in [7], we obtain that there is a subsequence of ( u n ) n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq179_HTML.gif, still denoted by ( u n ) n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq179_HTML.gif, and u λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq30_HTML.gif such that u n u λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq211_HTML.gif in C loc 1 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq212_HTML.gif. The function u λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq30_HTML.gif satisfies Eq. (1). Indeed, let [ x 1 , x 2 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq213_HTML.gif be an interval of and T n > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq214_HTML.gif such that [ x 1 , x 2 ] [ T n , T n ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq215_HTML.gif. By the above considerations, taking a limit as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq216_HTML.gif in the equation
( u n | u n | p 2 ) a ( x ) u n p 1 + λ b ( x ) u n q 1 = 0 , x [ x 1 , x 2 ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equar_HTML.gif
equivalent to
u n | u n | p 2 ( x ) = u n | u n | p 2 ( x 1 ) + x 1 x ( a ( t ) u n p 1 ( t ) λ b ( t ) u n q 1 ( t ) ) d t = 0 , x [ x 1 , x 2 ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equas_HTML.gif
we obtain
u λ | u λ | p 2 ( x ) = u λ | u λ | p 2 ( x 1 ) + x 1 x ( a ( t ) u λ p 1 ( t ) λ b ( t ) u λ q 1 ( t ) ) d t = 0 , x [ x 1 , x 2 ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equat_HTML.gif
and hence
( u λ | u λ | p 2 ) a ( x ) u λ p 1 + λ b ( x ) u λ q 1 = 0 , x [ x 1 , x 2 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equau_HTML.gif
Since x 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq87_HTML.gif and x 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq217_HTML.gif are arbitrary, u λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq30_HTML.gif is a solution of (1). Moreover, we have
( | u λ ( x ) | p + a ( x ) | u λ ( x ) | p ) d x < . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equ16_HTML.gif
(16)

It remains to show that u λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq30_HTML.gif is nonzero and u λ ( ± ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq218_HTML.gif and u λ ( ± ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq219_HTML.gif.

By Theorem 4, u n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq176_HTML.gif is an even function and attains its maximum at 0. Then by Eq. (1),
u n p 1 ( 0 ) ( a ( 0 ) + λ b ( 0 ) u n q p ( 0 ) ) 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equav_HTML.gif
By assumption (H)
u n ( 0 ) ( a ( 0 ) λ b ( 0 ) ) 1 / ( q p ) ( a λ B ) 1 / ( q p ) = C 3 > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equaw_HTML.gif
independently of n. Hence, passing to a limit as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq216_HTML.gif, we obtain
u λ ( 0 ) ( a λ B ) 1 / ( q p ) > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equax_HTML.gif

Note, that this implies max { u λ ( x ) : x R } = u λ ( 0 ) + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq220_HTML.gif as λ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq37_HTML.gif.

From (16) and Proposition 5, it follows
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equ17_HTML.gif
(17)

so u λ ( ± ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq218_HTML.gif.

Now, we will show that u λ ( ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq221_HTML.gif. The arguments for u λ ( ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq222_HTML.gif are similar.

If u λ ( ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq223_HTML.gif, there exist ε 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq224_HTML.gif and a monotone increasing sequence x k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq225_HTML.gif such that | u λ ( x k ) | ( 2 ε 1 ) 1 / ( p 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq226_HTML.gif. Then for x [ x k , x k + ε 1 K 2 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq227_HTML.gif,
| u λ ( x ) | p 1 = | φ p ( u λ ( x k ) ) + x k x φ p ( u λ ( t ) ) d t | | u λ ( x k ) | p 1 x k x k + ε 1 K 2 | φ p ( u λ ( t ) ) | d t 2 ε 1 ε 1 K 2 K 2 = ε 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equay_HTML.gif

which contradicts (16).

Moreover, u is an even function that attains its only maximum at 0, since the same holds for the functions u n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq176_HTML.gif. Arguing as in the proof of Theorem 4, we easily obtain that u ( x ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq228_HTML.gif if x > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq39_HTML.gif. □

Remark 2 A simplified method can be applied to the equations
u a ( x ) u | u | p 2 + λ b ( x ) u | u | q 2 = 0 , x R , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equaz_HTML.gif
under assumptions (H) and 2 p < q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq2_HTML.gif, λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq3_HTML.gif. Namely, first one looks for the even positive solutions u T , λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq28_HTML.gif of the problem
{ u a ( x ) φ p ( u ) + λ b ( x ) φ q ( u ) = 0 , x ( T , T ) , u ( T ) = u ( T ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equba_HTML.gif
considering the functional I T : H 0 1 ( T , T ) R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq229_HTML.gif
I T ( u ) = T T ( 1 2 u ( x ) 2 + 1 p a ( x ) | u ( x ) | p λ q b ( x ) ( u + ( x ) ) q ) d x , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equbb_HTML.gif
where H 0 1 ( T , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq230_HTML.gif is the Sobolev space of square integrable functions such that
u 2 = T T u ( x ) 2 d x < . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equbc_HTML.gif
Since H 0 1 ( T , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq230_HTML.gif is a Hilbert space, compactly embedded in C ( [ T , T ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq231_HTML.gif the proof of the (PS)-condition is easier. Similar considerations are made in [1] and [3]. Then, the even homoclinic solution u λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq30_HTML.gif is obtained as a C loc 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq232_HTML.gif limit of the sequence u T , λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq28_HTML.gif. Note that in this case, the even homoclinic solution u λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq30_HTML.gif of Eq. (3) satisfies
max { u λ ( x ) : x R } = u λ ( 0 ) ( a ( 0 ) λ b ( 0 ) ) 1 / ( q p ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Equbd_HTML.gif
and again u λ ( 0 ) + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq40_HTML.gif as λ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq37_HTML.gif. If a and b are constants, Eq. (3) is a conservative system and one can plot the phase curves ( v 2 ) 2 a | u | p p + λ b | u | q q = C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq44_HTML.gif in the phase plane ( u , v ) = ( u , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq45_HTML.gif. Consider the equation u u 3 + λ u 5 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq233_HTML.gif. The phase portrait in a ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq234_HTML.gif plane, for λ = 0.5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq235_HTML.gif in the rectangle { ( u , v ) : 2 u 2 , 1.25 v 1.25 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq236_HTML.gif, is plotted on Figure 2.
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_Fig2_HTML.jpg
Figure 2

Phase portrait of u u 3 + 0.5 u 5 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq237_HTML.gif , in [ 2 , 2 ] × [ 1.25 , 1.25 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-121/MediaObjects/13661_2012_Article_218_IEq238_HTML.gif .

Declarations

Acknowledgements

Dedicated to Professor Jean Mawhin on the occasion of his 70th anniversary.

The author thanks Prof. Alberto Cabada and Prof. Luis Sanchez for helpful remarks concerning Theorem 4. The author would like to thank the Department of Mathematics and Theoretical Informatics at the Technical University of Kosice, Slovakia, where the paper was prepared during his visit on the SAIA Fellowship programme. The author is thankful to the editor and anonymous referee for their comments and suggestions on the article.

Authors’ Affiliations

(1)
Department of Mathematical Analysis, University of Ruse

References

  1. Korman P, Lazer A: Homoclinic orbits for a class of symmetric hamiltonian systems. Electron. J. Differ. Equ. 1994., 1994: Article ID 1
  2. Grossinho MR, Sanchez L: A note on periodic solutions of some nonautonomous differential equations. Bull. Aust. Math. Soc. 1986, 34: 253-265. 10.1017/S000497270001011XMathSciNetView Article
  3. Grossinho MR, Minhos F, Tersian S: Positive homoclinic solutions for a class of second order differential equations. J. Math. Anal. Appl. 1999, 240: 163-173. 10.1006/jmaa.1999.6606MathSciNetView Article
  4. Austin G: Biomathematical model of aneurysm of the circle of Willis I: the Duffing equation and some approximate solutions. Math. Biosci. 1971, 11: 163-172. 10.1016/0025-5564(71)90015-0View Article
  5. Cronin J: Biomathematical model of aneurysm of the circle of Willis: a quantitative analysis of the differential equation of Austin. Math. Biosci. 1973, 16: 209-225. 10.1016/0025-5564(73)90031-XMathSciNetView Article
  6. Nieto JJ, Torres A: A nonlinear biomathematical model for the study of intracranial aneurysms. J. Neurol. Sci. 2000, 177: 18-23. 10.1016/S0022-510X(00)00315-4View Article
  7. Tang XH, Li X: Homoclinic solutions for ordinary p -Laplacian systems with a coercive potential. Nonlinear Anal. 2009, 71: 1124-1132. 10.1016/j.na.2008.11.027View Article
  8. Xu B, Tang C-L: Some existence results on periodic solutions of ordinary p -Laplacian systems. J. Math. Anal. Appl. 2007, 333: 1228-1236. 10.1016/j.jmaa.2006.11.051MathSciNetView Article
  9. Korman P, Ouyang T: Exact multiplicity results for two classes of boundary value problems. Differ. Integral Equ. 1993, 6(6):1507-1517.MathSciNet
  10. Gidas B, Ni WM, Nirenberg L: Symmetry and related properties via the maximum principle. Commun. Math. Phys. 1979, 68: 209-243. 10.1007/BF01221125MathSciNetView Article
  11. Reihel W, Walter W: Radial solutions of equations and inequalities involving the p -Laplacian. J. Inequal. Appl. 1997, 1: 47-71.MathSciNet
  12. Cabada A, Cid JA, Pouso RL: Positive solutions for a class of singular differential equations arising in diffusion processes. Dyn. Contin. Discrete Impuls. Syst. Ser. A, Math. Anal. 2005, 12: 329-342.MathSciNet
  13. Rabinowitz P: Homoclinic orbits for a class of Hamiltonian systems. Proc. R. Soc. Edinb. A 1990, 114: 33-38. 10.1017/S0308210500024240MathSciNetView Article
  14. Aizicovici S, Papageorgiou NS, Staicu V: Periodic solutions of nonlinear evolution inclusions in Banach spaces. J. Nonlinear Convex Anal. 2011, 7(2):163-177.MathSciNet
  15. Del Pino M, Drabek P, Manásevich R: The Fredholm alternative at the first eigenvalue for the one-dimensional p -Laplacian. J. Differ. Equ. 1999, 151: 386-419. 10.1006/jdeq.1998.3506View Article
  16. Ambrosetti A, Rabinowitz P: Dual variational methods in critical point theory and applications. J. Funct. Anal. 1973, 14: 349-381. 10.1016/0022-1236(73)90051-7MathSciNetView Article

Copyright

© Tersian; licensee Springer 2012

This article is published under license to BioMed Central Ltd. 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.