Open Access

Ground state and mountain pass solutions for discrete p ( ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq1_HTML.gif-Laplacian

Boundary Value Problems20122012:104

DOI: 10.1186/1687-2770-2012-104

Received: 25 July 2012

Accepted: 5 September 2012

Published: 19 September 2012

Abstract

In this paper we study the existence of solutions for discrete p ( ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq2_HTML.gif-Laplacian equations subjected to a potential type boundary condition. Our approach relies on Szulkin’s critical point theory and enables us to obtain the existence of ground state as well as mountain pass type solutions.

MSC: 39A12, 39A70, 49J40, 65Q10.

Keywords

discrete p ( ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq2_HTML.gif-Laplacian operator variational methods critical point Palais-Smale condition Mountain Pass Theorem

1 Introduction

Let T be a positive integer, p : [ 0 , T ] ( 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq3_HTML.gif and h p ( k ) : R R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq4_HTML.gif be defined by h p ( k ) ( x ) = | x | p ( k ) 2 x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq5_HTML.gif for all x R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq6_HTML.gif and k [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq7_HTML.gif. Here and below, for a , b N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq8_HTML.gif with a < b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq9_HTML.gif, we use the notation [ a , b ] : = { a , a + 1 , , b } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq10_HTML.gif.

This paper is concerned with the existence of solutions for equations of the type
Δ p ( k 1 ) x ( k 1 ) = f ( k , x ( k ) ) , ( ) k [ 1 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ1_HTML.gif
(1.1)
subjected to the potential boundary condition
( h p ( 0 ) ( Δ x ( 0 ) ) , h p ( T ) ( Δ x ( T ) ) ) j ( x ( 0 ) , x ( T + 1 ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ2_HTML.gif
(1.2)
where Δ x ( k ) = x ( k + 1 ) x ( k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq11_HTML.gif is the forward difference operator and Δ p ( ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq12_HTML.gif stands for the discrete p ( ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq13_HTML.gif-Laplacian operator, that is,
Δ p ( k 1 ) x ( k 1 ) : = Δ ( h p ( k 1 ) ( Δ x ( k 1 ) ) ) = h p ( k ) ( Δ x ( k ) ) h p ( k 1 ) ( Δ x ( k 1 ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equa_HTML.gif
Here and hereafter, f : [ 1 , T ] × R R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq14_HTML.gif is a continuous function, while j : R × R ( , + ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq15_HTML.gif is convex, proper (i.e., D ( j ) : = { z R × R : j ( z ) < + } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq16_HTML.gif), lower semicontinuous (in short, l.s.c.) and ∂j denotes the subdifferential of j. Recall, for z R × R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq17_HTML.gif, the set j ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq18_HTML.gif is defined by
j ( z ) = { ζ R × R : j ( ξ ) j ( z ) ( ζ | ξ z ) , ( ) ξ R × R } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ3_HTML.gif
(1.3)

where ( | ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq19_HTML.gif stands for the usual inner product in R × R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq20_HTML.gif.

It should be noticed that the boundary condition (1.2) recovers the classical ones. For instance, denoting by I K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq21_HTML.gif the indicator function of a closed, nonempty and convex set K R × R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq22_HTML.gif, the Dirichlet and Neumann boundary conditions are obtained by choosing j = I K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq23_HTML.gif with K = { ( 0 , 0 ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq24_HTML.gif and K = R × R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq25_HTML.gif, respectively. If p is T-periodic, taking K = { ( x , x ) , x R } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq26_HTML.gif ( K = { ( x , x ) , x R } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq27_HTML.gif) and j = I K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq23_HTML.gif, we get the periodic (antiperiodic) conditions. For other choices of j yielding various boundary conditions, we refer the reader to Gasinski and Papageorgiou [1] and Jebelean [2].

The study of boundary value problems with a discrete p-Laplacian using variational approaches has captured attention in the last years. Most of the papers deal with classical boundary conditions such as Dirichlet (see, e.g., Agarwal et al. [3], Cabada et al. [4]), Neumann (Candito and D’Agui [5], Tian and Ge [6]) and periodic (He and Chen [7], Jebelean and Şerban [8]). Also, we note the recent paper of Mawhin [9] where variational techniques are employed to obtain the existence of periodic solutions for systems involving a general discrete ϕ-Laplacian operator.

Boundary value problems with the discrete p ( ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq13_HTML.gif-Laplacian subjected to Dirichlet, Neumann or periodic boundary conditions were studied in recent time by Bereanu et al. [10], Galewski and Glab [11, 12], Guiro et al. [13], Koné and Ouaro [14], Mashiyev et al. [15], Mihăilescu et al. [16, 17].

Here, we use a variational approach to obtain ground state and mountain pass solutions for problem (1.1), (1.2). In this view, we employ some ideas originated in Jebelean and Moroşanu [18] (also see Jebelean [2]) combined with specific technicalities due to the discrete and anisotropic character of the problem. The main existence results are Theorem 3.1 and Theorem 4.2. These recover and generalize the similar ones for p= constant obtained in [19].

The rest of the paper is organized as follows. The functional framework and the variational approach of problem (1.1), (1.2) are presented in Section 2. In Section 3, we obtain the existence of ground state solutions, while Section 4 is devoted to the existence of mountain pass type solutions. An example of application is given in Section 5.

2 The functional framework

Our approach for the boundary value problem (1.1), (1.2) relies on the critical point theory developed by Szulkin [20]. With this aim, we introduce the space
X : = { x : [ 0 , T + 1 ] R } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equb_HTML.gif
which will be considered with the Luxemburg norm
x η , p ( ) = inf { ν > 0 : k = 1 T + 1 1 p ( k 1 ) | Δ x ( k 1 ) ν | p ( k 1 ) + η k = 1 T 1 p ( k ) | x ( k ) ν | p ( k ) 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equc_HTML.gif

for some η > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq28_HTML.gif. Also, we shall make use of the usual sup-norm x = max k [ 0 , T + 1 ] | x ( k ) | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq29_HTML.gif.

Next, let φ : X R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq30_HTML.gif be defined by
φ ( x ) = k = 1 T + 1 1 p ( k 1 ) | Δ x ( k 1 ) | p ( k 1 ) , ( ) x X . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ4_HTML.gif
(2.1)
Standard arguments show that φ is convex, of class C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq31_HTML.gif. Using the summation by parts formula (see, e.g., [8, 19]), one obtains that its derivative is given by
φ ( x ) , y = k = 1 T + 1 h p ( k 1 ) ( Δ x ( k 1 ) ) Δ y ( k 1 ) , ( ) x , y X . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ5_HTML.gif
(2.2)
By means of j, we introduce the functional J : X ( , + ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq32_HTML.gif given by
J ( x ) = j ( x ( 0 ) , x ( T + 1 ) ) , ( ) x X . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ6_HTML.gif
(2.3)
Note that, as j is proper, convex and l.s.c., the same properties hold true for J. Then setting
ψ = φ + J , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ7_HTML.gif
(2.4)

it is clear that ψ is proper, convex and l.s.c. on X.

Further, we define F : [ 1 , T ] × R R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq33_HTML.gif by
F ( k , t ) = 0 t f ( k , τ ) d τ , ( ) k [ 1 , T ] , ( ) t R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equd_HTML.gif
and
Φ ( x ) = k = 1 T F ( k , x ( k ) ) , ( ) x X . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ8_HTML.gif
(2.5)
It is a simple matter to check that Φ C 1 ( X , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq34_HTML.gif and
Φ ( x ) , y = k = 1 T f ( k , x ( k ) ) y ( k ) , ( ) x , y X . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ9_HTML.gif
(2.6)
The energy functional associated to problem (1.1), (1.2) is
I = Φ + ψ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Eque_HTML.gif

with ψ in (2.4) and Φ given by (2.5).

Proposition 2.1 If x X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq35_HTML.gif is a critical point of the functional I https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq36_HTML.gif in the sense that
Φ ( x ) , y x + ψ ( y ) ψ ( x ) 0 , ( ) y X , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ10_HTML.gif
(2.7)

then x is a solution of problem (1.1), (1.2).

Proof In (2.7), we take y = x + s w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq37_HTML.gif, s > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq38_HTML.gif; then dividing by s and letting s 0 + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq39_HTML.gif, we get
Φ ( x ) , w + φ ( x ) , w + J ( x ; w ) 0 , ( ) w X , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equf_HTML.gif
where J ( x ; w ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq40_HTML.gif is the directional derivative of the convex function J at x in the direction of w. By virtue of (2.3), the above inequality becomes
Φ ( x ) , w + φ ( x ) , w + j ( ( x ( 0 ) , x ( T + 1 ) ) ; ( w ( 0 ) , w ( T + 1 ) ) ) 0 , ( ) w X . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equg_HTML.gif
Using (2.2), (2.6) and the summation by parts formula, a straightforward computation shows that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ11_HTML.gif
(2.8)
for all w X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq41_HTML.gif. Thus, we infer
k = 1 T ( Δ p ( k 1 ) x ( k 1 ) + f ( k , x ( k ) ) ) w ( k ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equh_HTML.gif
for all w X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq41_HTML.gif with w ( 0 ) = w ( T + 1 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq42_HTML.gif. This implies that
Δ p ( k 1 ) x ( k 1 ) = f ( k , x ( k ) ) , ( ) k [ 1 , T ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ12_HTML.gif
(2.9)
To prove that x satisfies condition (1.2), we multiply the equality (2.9) by w ( k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq43_HTML.gif. Then summing from 1 to T and using (2.8), one obtains
j ( ( x ( 0 ) , x ( T + 1 ) ) ; ( w ( 0 ) , w ( T + 1 ) ) ) h p ( T ) ( Δ x ( T ) ) w ( T + 1 ) + h p ( 0 ) ( Δ x ( 0 ) ) w ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equi_HTML.gif
for all w X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq41_HTML.gif. Taking w X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq41_HTML.gif with w ( 0 ) = u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq44_HTML.gif and w ( T + 1 ) = v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq45_HTML.gif, where u , v R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq46_HTML.gif are arbitrarily chosen, we have
j ( ( x ( 0 ) , x ( T + 1 ) ) ; ( u , v ) ) h p ( 0 ) ( Δ x ( 0 ) ) u h p ( T ) ( Δ x ( T ) ) v , ( ) u , v R , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equj_HTML.gif
which, by a standard result from convex analysis, means that
( h p ( 0 ) ( Δ x ( 0 ) ) , h p ( T ) ( Δ x ( T ) ) ) j ( x ( 0 ) , x ( T + 1 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equk_HTML.gif

and the proof is complete. □

From now on, we will use the following notations:
p = min k [ 0 , T ] p ( k ) , p + = max k [ 0 , T ] p ( k ) and p ̲ = min k [ 1 , T ] p ( k ) , p ¯ = max k [ 1 , T ] p ( k ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equl_HTML.gif
Remark 2.2 It is easy to check that for all x X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq35_HTML.gif and any η > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq28_HTML.gif, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ13_HTML.gif
(2.10)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ14_HTML.gif
(2.11)

3 Ground state solutions

We begin by a result which states that the energy functional I https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq36_HTML.gif has a minimum point in X provided that the potential of the nonlinearity f lies asymptotically on the left of the first eigenvalue like constant
λ 1 : = inf { k = 1 T + 1 | Δ x ( k 1 ) | p ( k 1 ) p ( k 1 ) k = 1 T | x ( k ) | p ( k ) p ( k ) : x X { 0 } , ( x ( 0 ) , x ( T + 1 ) ) D ( j ) } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ15_HTML.gif
(3.1)
Theorem 3.1 If
lim sup | t | p ( k ) F ( k , t ) | t | p ( k ) < λ 1 , ( ) k [ 1 , T ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ16_HTML.gif
(3.2)

then problem (1.1), (1.2) has at least one solution which minimizes I https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq36_HTML.gif on X.

Proof By the continuity of Φ and the lower semicontinuity of ψ, we have that the functional I https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq36_HTML.gif is sequentially l.s.c. on X. It remains to prove that I https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq36_HTML.gif is coercive on X. Then, by the direct method in calculus of variations, I https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq36_HTML.gif is bounded from below and attains its infimum at some x X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq35_HTML.gif, which, by virtue of ([20], Proposition 1.1) and Proposition 2.1, is a solution of problem (1.1), (1.2).

From (3.2) there are constants σ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq47_HTML.gif and ρ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq48_HTML.gif such that
F ( k , t ) λ 1 σ p ( k ) | t | p ( k ) , ( ) k [ 1 , T ] , ( ) t R  with  | t | > ρ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equm_HTML.gif
If λ 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq49_HTML.gif, we may assume that σ < λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq50_HTML.gif. On the other hand, by the continuity of F, there is a constant M ρ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq51_HTML.gif such that
| F ( k , t ) | M ρ , ( ) k [ 1 , T ] , ( ) t R  with  | t | ρ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equn_HTML.gif
Hence, we infer
F ( k , t ) M ρ + | λ 1 σ | p ( k ) ρ p ( k ) + λ 1 σ p ( k ) | t | p ( k ) , ( ) k [ 1 , T ] , ( ) t R . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equo_HTML.gif
To prove the coercivity of I https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq36_HTML.gif, from the above inequality, we obtain
I ( x ) φ ( x ) M ρ T | λ 1 σ | k = 1 T 1 p ( k ) ρ p ( k ) ( λ 1 σ ) k = 1 T 1 p ( k ) | x ( k ) | p ( k ) + J ( x ) φ ( x ) M ρ T | λ 1 σ | p ( ρ p + ρ p + ) T ( λ 1 σ ) k = 1 T 1 p ( k ) | x ( k ) | p ( k ) + J ( x ) = φ ( x ) ( λ 1 σ ) k = 1 T 1 p ( k ) | x ( k ) | p ( k ) C 1 + J ( x ) , ( ) x X , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equp_HTML.gif

where C 1 = M ρ T + | λ 1 σ | p ( ρ p + ρ p + ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq52_HTML.gif.

If λ 1 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq53_HTML.gif, using (2.11) from Remark 2.2, we have
I ( x ) φ ( x ) + σ k = 1 T 1 p ( k ) | x ( k ) | p ( k ) C 1 + J ( x ) x σ , p ( ) p C 1 + J ( x ) , ( ) x X , x σ , p ( ) > 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ17_HTML.gif
(3.3)
In the case λ 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq49_HTML.gif, by virtue of (2.11) and (3.1), for x λ 1 , p ( ) > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq54_HTML.gif, one obtains
x λ 1 , p ( ) p φ ( x ) + λ 1 k = 1 T 1 p ( k ) | x ( k ) | p ( k ) 2 φ ( x ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ18_HTML.gif
(3.4)
which, using again (3.1), implies
I ( x ) φ ( x ) + ( σ λ 1 ) k = 1 T + 1 1 p ( k 1 ) | Δ x ( k 1 ) | p ( k 1 ) λ 1 C 1 + J ( x ) = σ λ 1 φ ( x ) C 1 + J ( x ) σ 2 λ 1 x λ 1 , p ( ) p C 1 + J ( x ) , ( ) x D ( J ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ19_HTML.gif
(3.5)
In both cases, by virtue of (3.3) and (3.5), there exist constants η , C 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq55_HTML.gif such that
I ( x ) C 2 x η , p ( ) p C 1 + J ( x ) , ( ) x D ( J ) , x η , p ( ) > 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equq_HTML.gif
On the other hand, as j is convex and l.s.c., it is bounded from below by an affine functional. Therefore, on account of (2.3), there are positive constants k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq56_HTML.gif, k 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq57_HTML.gif, k 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq58_HTML.gif such that
I ( x ) C 2 x η , p ( ) p C 1 k 1 | x ( 0 ) | k 2 | x ( T + 1 ) | k 3 C 2 x η , p ( ) p C 3 x C 4 , ( ) x D ( J ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equr_HTML.gif
with C 3 = k 1 + k 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq59_HTML.gif and C 4 = C 1 + k 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq60_HTML.gif. Since any norm on X is equivalent to η , p ( ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq61_HTML.gif, there exists C 5 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq62_HTML.gif such that
I ( x ) C 2 x η , p ( ) p C 5 x η , p ( ) C 4 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equs_HTML.gif
Consequently,
I ( x ) + , as  x η , p ( ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equt_HTML.gif

meaning that I https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq36_HTML.gif is coercive on ( X , η , p ( ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq63_HTML.gif and the proof is complete. □

In order to give an application of Theorem 3.1, we consider the problem
{ Δ p ( k 1 ) x ( k 1 ) = λ g ( k , x ( k ) ) , ( ) k [ 1 , T ] , ( h p ( 0 ) ( Δ x ( 0 ) ) , h p ( T ) ( Δ x ( T ) ) ) j ( x ( 0 ) , x ( T + 1 ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ20_HTML.gif
(3.6)

where g : [ 1 , T ] × R R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq64_HTML.gif is a continuous function and λ is a positive parameter.

Corollary 3.2 Assume that λ 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq49_HTML.gif and g satisfies the growth condition
| g ( k , t ) | a | t | q ( k ) + b , ( ) k [ 1 , T ] , ( ) t R , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ21_HTML.gif
(3.7)

where a > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq65_HTML.gif, b R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq66_HTML.gif are constants and q : [ 1 , T ] [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq67_HTML.gif. The following hold true:

(i) if p ̲ > q ¯ + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq68_HTML.gif, then problem (3.6) has a solution for any λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq69_HTML.gif;

(ii) if p ̲ = q ¯ + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq70_HTML.gif, then there is some λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq71_HTML.gif such that for any λ ( 0 , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq72_HTML.gif, problem (3.6) has a solution.

Proof We apply Theorem 3.1 with f ( k , t ) = λ g ( k , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq73_HTML.gif for all k [ 1 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq74_HTML.gif and t R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq75_HTML.gif. From (3.7), we obtain
| F ( k , t ) | = | 0 t f ( k , τ ) d τ | λ 0 | t | ( a τ q ( k ) + b ) d τ = λ a | t | q ( k ) + 1 q ( k ) + 1 + λ | t | b λ a q ̲ + 1 | t | q ( k ) + 1 + λ | t | b , ( ) k [ 1 , T ] , ( ) t R . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equu_HTML.gif
Thus, we deduce
p ( k ) F ( k , t ) | t | p ( k ) p ¯ λ a q ̲ + 1 | t | q ( k ) + 1 | t | p ( k ) + p ¯ λ b | t | | t | p ( k ) p ¯ λ a q ̲ + 1 | t | q ¯ + 1 | t | p ̲ + p ¯ λ b | t | | t | p ̲ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equv_HTML.gif
for all k [ 1 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq74_HTML.gif and t R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq75_HTML.gif with | t | > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq76_HTML.gif. So, if p ̲ > q ¯ + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq68_HTML.gif, then
lim sup | t | p ( k ) F ( k , t ) | t | p ( k ) 0 < λ 1 , ( ) k [ 1 , T ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equw_HTML.gif
Also, if p ̲ = q ¯ + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq70_HTML.gif, setting
λ = λ 1 ( q ̲ + 1 ) a p ¯ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ22_HTML.gif
(3.8)

it is easy to see that condition (3.2) is fulfilled for any λ ( 0 , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq72_HTML.gif. □

Remark 3.3

(i) Note that a valid λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq77_HTML.gif in Corollary 3.2(ii) is given by formula (3.8).

(ii) Theorem 5 in [11] is an immediate consequence of Corollary 3.2 with j = I K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq23_HTML.gif, K = { ( 0 , 0 ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq24_HTML.gif.

4 Mountain pass type solutions

In this section, we deal with the existence of nontrivial solutions for the equation
Δ p ( k 1 ) x ( k 1 ) + r ( k ) h p ( k ) ( x ( k ) ) = f ( k , x ( k ) ) , ( ) k [ 1 , T ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ23_HTML.gif
(4.1)

associated with the potential boundary condition (1.2). Here, f and j are as in the case of the previous problem (1.1), (1.2) and r : [ 1 , T ] [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq78_HTML.gif is a given function. The main tool in obtaining such a result will be the Mountain Pass Theorem [20].

To treat problem (4.1), (1.2), instead of φ, there will be φ r : X R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq79_HTML.gif defined by
φ r ( x ) = k = 1 T + 1 1 p ( k 1 ) | Δ x ( k 1 ) | p ( k 1 ) + k = 1 T r ( k ) p ( k ) | x ( k ) | p ( k ) , ( ) x X , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ24_HTML.gif
(4.2)
which is convex, of class C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq31_HTML.gif on X, and its derivative is given by
φ r ( x ) , y = k = 1 T + 1 h p ( k 1 ) ( Δ x ( k 1 ) ) Δ y ( k 1 ) + k = 1 T r ( k ) h p ( k ) ( x ( k ) ) y ( k ) , ( ) x , y X . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ25_HTML.gif
(4.3)
Then, setting ψ r = φ r + J https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq80_HTML.gif, we define
I r = ψ r + Φ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ26_HTML.gif
(4.4)

with J given by (2.3) and Φ in (2.5).

By means of λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq81_HTML.gif in (3.1), we define the constants
η ̲ : = λ 1 + r ̲ and η ¯ : = λ 1 + r ¯ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ27_HTML.gif
(4.5)
Lemma 4.1 If η ̲ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq82_HTML.gif and there exist constants θ > p + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq83_HTML.gif and C 1 , ρ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq84_HTML.gif such that
j ( z ; z ) θ j ( z ) + C 1 , ( ) z D ( j ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ28_HTML.gif
(4.6)
and
θ F ( k , t ) t f ( k , t ) , ( ) k [ 1 , T ] , ( ) t R with | t | > ρ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ29_HTML.gif
(4.7)
then the functional I r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq85_HTML.gif defined in (4.4) satisfies the Palais-Smale condition ((PS) condition for short) on ( X , η ̲ , p ( ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq86_HTML.gif, i.e., every sequence { x n } X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq87_HTML.gif for which I r ( x n ) c R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq88_HTML.gif and
Φ ( x n ) , y x n + ψ r ( y ) ψ r ( x n ) ε n y x n η ̲ , p ( ) , ( ) y X , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ30_HTML.gif
(4.8)

where ε n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq89_HTML.gif, possesses a convergent subsequence.

Proof Let { x n } X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq87_HTML.gif be a sequence for which I r ( x n ) c R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq88_HTML.gif and (4.8) holds true with ε n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq89_HTML.gif. Since X is finite dimensional, it is sufficient to prove that { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq90_HTML.gif is bounded. In order to show this, we may assume that { x n } D ( I r ) = D ( J ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq91_HTML.gif and x n η ̲ , p ( ) > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq92_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq93_HTML.gif. By virtue of (2.11), (3.1) and (4.5), we get
x n η ̲ , p ( ) p k = 1 T + 1 | Δ x n ( k 1 ) | p ( k 1 ) p ( k 1 ) + ( λ 1 + r ̲ ) k = 1 T 1 p ( k ) | x n ( k ) | p ( k ) 2 p ( k = 1 T + 1 | Δ x n ( k 1 ) | p ( k 1 ) + k = 1 T r ( k ) | x n ( k ) | p ( k ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ31_HTML.gif
(4.9)
From (2.3) and (4.6), it follows
J ( y ) 1 θ J ( y ; y ) C 2 , ( ) y D ( J ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ32_HTML.gif
(4.10)
with C 2 = C 1 / θ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq94_HTML.gif. Using (4.7) we deduce that, for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq93_HTML.gif, it holds
Φ ( x n ) + 1 θ Φ ( x n ) , x n = 1 θ k = 1 T [ θ F ( k , x n ( k ) ) x n ( k ) f ( k , x n ( k ) ) ] 1 θ | x n ( k ) | ρ [ θ F ( k , x n ( k ) ) x n ( k ) f ( k , x n ( k ) ) ] 1 θ k = 1 T max | x | ρ | θ F ( k , x ) x f ( k , x ) | = : C 3 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ33_HTML.gif
(4.11)
Clearly, there is a constant C 4 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq95_HTML.gif, such that
| I r ( x n ) | C 4 , ( ) n N . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ34_HTML.gif
(4.12)
Further, setting y = x n + s x n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq96_HTML.gif in (4.8), dividing by s > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq38_HTML.gif and then letting s 0 + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq39_HTML.gif, we obtain
Φ ( x n ) , x n + φ r ( x n ) , x n + J ( x n ; x n ) ε n x n η ̲ , p ( ) , ( ) n N . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ35_HTML.gif
(4.13)
Using (4.12) and (4.13), we deduce that
C 4 + ε n θ x n η ̲ , p ( ) Φ ( x n ) + φ r ( x n ) + J ( x n ) + ε n θ x n η ̲ , p ( ) Φ ( x n ) 1 θ Φ ( x n ) , x n + φ r ( x n ) 1 θ φ r ( x n ) , x n + J ( x n ) 1 θ J ( x n ; x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equx_HTML.gif
and by virtue of (4.10), (4.11), (4.3) and (4.9), we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equy_HTML.gif

Since θ > p + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq83_HTML.gif, we infer that { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq90_HTML.gif is bounded and the proof is complete. □

Now, we can state the following result of Ambrosetti-Rabinowitz type [21].

Theorem 4.2 Assume that η ̲ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq82_HTML.gif and, in addition,

(i) ( 0 , 0 ) j ( 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq97_HTML.gif;

(ii) lim sup | t | 0 p ( k ) F ( k , t ) | t | p ( k ) < η ̲ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq98_HTML.gif, ( ) k [ 1 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq99_HTML.gif;

(iii) there are constants θ > p + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq83_HTML.gif and C 1 , ρ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq84_HTML.gif such that (4.6) holds true and
0 < θ F ( k , t ) t f ( k , t ) , ( ) k [ 1 , T ] , ( ) t R with | t | > ρ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ36_HTML.gif
(4.14)

Then, problem (4.1), (1.2) has a nontrivial solution.

Proof Without loss of generality, we may assume that
j ( 0 , 0 ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ37_HTML.gif
(4.15)
which implies that I r ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq100_HTML.gif. From (i), (2.3) and (4.15), we have
J ( x ) J ( 0 ) = 0 , ( ) x D ( J ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ38_HTML.gif
(4.16)

From Lemma 4.1 and (iii), the functional I r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq85_HTML.gif satisfies the (PS) condition on ( X , η ̲ , p ( ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq101_HTML.gif.

Next, we shall prove that I r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq85_HTML.gif has a ‘mountain pass’ geometry:

(a) there exist α , μ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq102_HTML.gif such that I r ( x ) α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq103_HTML.gif if x η ̲ , p ( ) = μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq104_HTML.gif;

(b) I r ( e ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq105_HTML.gif for some e X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq106_HTML.gif with e η ̲ , p ( ) > μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq107_HTML.gif.

By the equivalence of the norms on X, there is some C 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq108_HTML.gif such that
x C 2 x η ̲ , p ( ) , ( ) x X . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ39_HTML.gif
(4.17)
Using (ii) we can find constants σ ( 0 , η ̲ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq109_HTML.gif and μ ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq110_HTML.gif such that
F ( k , t ) η ̲ σ p ( k ) | t | p ( k ) , ( ) k [ 1 , T ] , ( ) t R  with  | t | μ C 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ40_HTML.gif
(4.18)
Let x X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq35_HTML.gif, with x η ̲ , p ( ) μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq111_HTML.gif, be arbitrarily chosen. From (4.17) and (4.18), we have
F ( k , x ( k ) ) η ̲ σ p ( k ) | x ( k ) | p ( k ) , ( ) k [ 1 , T ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equz_HTML.gif
which implies
Φ ( x ) ( η ̲ σ ) k = 1 T 1 p ( k ) | x ( k ) | p ( k ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equaa_HTML.gif
Now, using (2.10) and (3.1), we get
x η ̲ , p ( ) p + k = 1 T + 1 | Δ x ( k 1 ) | p ( k 1 ) p ( k 1 ) + ( λ 1 + r ̲ ) k = 1 T 1 p ( k ) | x ( k ) | p ( k ) 2 φ r ( x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ41_HTML.gif
(4.19)
By virtue of (3.1), (4.5), (4.2) and (4.19), we deduce
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equab_HTML.gif
On account of (4.16), we infer that
I r ( x ) = Φ ( x ) + φ r ( x ) + J ( x ) α , if  x η ̲ , p ( ) = μ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ42_HTML.gif
(4.20)

with σ μ p + 2 η ̲ = : α > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq112_HTML.gif, and condition (a) is fulfilled.

Our next task is to prove that I r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq85_HTML.gif satisfies condition (b). To this end, let us first observe that, by virtue of (4.14), there exist a 1 , a 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq113_HTML.gif such that
F ( k , t ) a 1 | t | θ a 2 , ( ) k [ 1 , T ] , ( ) t R . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ43_HTML.gif
(4.21)
Let x 0 X { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq114_HTML.gif be such that x 0 ( 0 ) = x 0 ( T + 1 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq115_HTML.gif and x 0 η ¯ , p ( ) > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq116_HTML.gif. Using (2.11) and (4.5), one obtains
x 0 η ¯ , p ( ) p + k = 1 T + 1 | Δ x 0 ( k 1 ) | p ( k 1 ) p ( k 1 ) + ( λ 1 + r ¯ ) k = 1 T 1 p ( k ) | x 0 ( k ) | p ( k ) φ r ( x 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ44_HTML.gif
(4.22)
From (4.15), we have that
J ( s x 0 ) = 0 , ( ) s R , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equac_HTML.gif
which, together with (4.21) and (4.22) for any s 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq117_HTML.gif, gives
I r ( s x 0 ) s p + φ r ( x 0 ) + Φ ( s x 0 ) s p + x 0 η ¯ , p ( ) p + s θ a 1 k = 1 T | x 0 ( k ) | θ + a 2 T , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equad_HTML.gif

as s + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq118_HTML.gif because θ > p + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq83_HTML.gif. Hence, we can choose s 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq119_HTML.gif large enough to satisfy I r ( s 0 x 0 ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq120_HTML.gif and s 0 x 0 η ̲ , p ( ) > μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq121_HTML.gif, with μ entering in (4.20). This means that condition (b) is satisfied with e = s 0 x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq122_HTML.gif. □

5 An application

In this section, we show how Theorem 4.2 can be applied to derive the existence of nontrivial solutions for equation (4.1) associated with some concrete boundary conditions.

Let g : R × R R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq123_HTML.gif be a convex and Gâteaux differentiable function with d g ( 0 , 0 ) = ( 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq124_HTML.gif, where dg denotes the differential of g. Also, given a nonempty closed convex cone K R × R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq125_HTML.gif, we denote by N K ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq126_HTML.gif the normal cone to K at z K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq127_HTML.gif, i.e.,
N K ( z ) = { ζ R × R : ( ζ | ξ z ) 0 , ( ) ξ K } , ( ) z K . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equae_HTML.gif
The equation (4.1) is considered to be associated with the boundary conditions
( x ( 0 ) , x ( T + 1 ) ) K , ( h p ( 0 ) ( Δ x ( 0 ) ) , h p ( T ) ( Δ x ( T ) ) ) d g ( x ( 0 ) , x ( T + 1 ) ) N K ( x ( 0 ) , x ( T + 1 ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ45_HTML.gif
(5.1)
We set
η ̲ K : = r ̲ + inf { k = 1 T + 1 1 p ( k 1 ) | Δ x ( k 1 ) | p ( k 1 ) k = 1 T 1 p ( k ) | x ( k ) | p ( k ) : x X { 0 } , ( x ( 0 ) , x ( T + 1 ) ) K } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equaf_HTML.gif

Theorem 5.1 If f : [ 1 , T ] × R R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq128_HTML.gif is continuous, η ̲ K > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq129_HTML.gif and, in addition, we assume that

(i) lim sup | t | 0 p ( k ) F ( k , t ) | t | p ( k ) < η ̲ K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq130_HTML.gif, ( ) k [ 1 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq99_HTML.gif;

(ii) there are constants θ > p + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq83_HTML.gif and C 1 , ρ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq84_HTML.gif such that (4.14) holds true and
d g ( z ) , z θ g ( z ) + C 1 , ( ) z K , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equ46_HTML.gif
(5.2)

then problem (4.1), (5.1) has a nontrivial solution.

Proof Since N K ( z ) = I K ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq131_HTML.gif for all z K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq127_HTML.gif, Theorem 4.2 applies with j ( z ) = g ( z ) + I K ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq132_HTML.gif, ( ) z R × R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq133_HTML.gif. □

Remark 5.2 Conditions (5.1) allow various possible choices of g and K, which, among others, recover classical boundary conditions. For instance, if g = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq134_HTML.gif, then the homogeneous boundary conditions
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equag_HTML.gif
are obtained by choosing K = { ( 0 , 0 ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq24_HTML.gif, respectively K = R × R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq25_HTML.gif. If, in addition, p is T-periodic, then taking K = { ( x , x ) , x R } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq26_HTML.gif and K = { ( x , x ) , x R } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq135_HTML.gif, we get
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equah_HTML.gif
respectively. If the T-periodicity condition is not assumed, then we only have
h p ( 0 ) ( Δ x ( 0 ) ) = h p ( T ) ( Δ x ( T ) ) and h p ( 0 ) ( Δ x ( 0 ) ) = h p ( T ) ( Δ x ( T ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equai_HTML.gif

instead of Δ x ( 0 ) = Δ x ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq136_HTML.gif and Δ x ( 0 ) = Δ x ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq137_HTML.gif, respectively. As g = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq134_HTML.gif, in these four cases, condition (5.2) is automatically satisfied with any θ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq138_HTML.gif and C 1 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq139_HTML.gif.

Also, if α , β > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq140_HTML.gif are given, then with g defined by
g ( z ) : = 1 p ( 0 ) | z 1 | p ( 0 ) α p ( 0 ) 1 + 1 p ( T ) | z 2 | p ( T ) β p ( T ) 1 , ( ) z = ( z 1 , z 2 ) R × R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equaj_HTML.gif
and K = R × R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq25_HTML.gif, we deduce the Sturm-Liouville type boundary conditions
x ( 0 ) α Δ x ( 0 ) = 0 , x ( T + 1 ) + β Δ x ( T ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_Equak_HTML.gif

In this case, (5.2) is fulfilled with any θ min { p ( 0 ) , p ( T ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq141_HTML.gif and C 1 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq139_HTML.gif.

Therefore, sufficient conditions ensuring the existence of nontrivial solutions of (4.1) subjected to one of the above boundary conditions can be easily stated by means of Theorem 5.1.

Remark 5.3 It is worth pointing out that in the cases of Dirichlet and antiperiodic boundary conditions, r ̲ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq142_HTML.gif is allowed to be =0, and hence, r may be ≥0 on [ 1 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq143_HTML.gif; while in the Neumann, periodic and Sturm-Liouville cases, r ̲ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq142_HTML.gif must be >0, meaning r > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq144_HTML.gif on [ 1 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-104/MediaObjects/13661_2012_Article_213_IEq143_HTML.gif.

Declarations

Acknowledgements

Dedicated to Professor Jean Mawhin for his 70th anniversary.

The research of CŞ was supported by the strategic grant POSDRU/CPP107/DMI1.5/S/78421, Project ID 78421 (2010), co-financed by the European Social Fund - Investing in People, within the Sectoral Operational Programme Human Resources Development 2007-2013. Also, the support for CB and PJ from the grant TE-PN-II-RU-TE-2011-3-0157 (CNCS-Romania) is gratefully acknowledged.

Authors’ Affiliations

(1)
Institute of Mathematics “Simion Stoilow”, Romanian Academy
(2)
Department of Mathematics, West University of Timişoara

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