Journal of Inequalities and Applications
VolumeΒ 2009Β (2009), Article IDΒ 926518, 22 pages
doi:10.1155/2009/926518
Research Article

Existence and Asymptotic Behavior of Solutions for Weighted 𝑝 ( 𝑑 ) -Laplacian System Multipoint Boundary Value Problems in Half Line

Zhimei Qiu,1Β Qihu Zhang,1,2Β and Yan Wang1

1School of Mathematical Science, Xuzhou Normal University, Xuzhou, Jiangsu 221116, China
2Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, Henan 450002, China

Received 5 January 2009; Accepted 20 June 2009

Academic Editor: AlbertoΒ Cabada

Copyright Β© 2009 Zhimei Qiu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper investigates the existence and asymptotic behavior of solutions for weighted 𝑝 ( 𝑑 ) -Laplacian system multipoint boundary value problems in half line. When the nonlinearity term 𝑓 ( 𝑑 , β‹… , β‹… ) satisfies sub-( 𝑝 βˆ’ βˆ’ 1 ) growth condition or general growth condition, we give the existence of solutions via Leray-Schauder degree.

1. Introduction

In this paper, we consider the existence and asymptotic behavior of solutions for the following weighted 𝑝 ( 𝑑 ) -Laplacian system: βˆ’ Ξ” 𝑝 ( 𝑑 ) ξ€· 𝑒 + 𝛿 𝑓 𝑑 , 𝑒 , ( 𝑀 ( 𝑑 ) ) 1 / ( 𝑝 ( 𝑑 ) βˆ’ 1 ) 𝑒 ξ…ž ξ€Έ = 0 , 𝑑 ∈ ( 0 , + ∞ ) , ( 1 . 1 ) 𝑒 ( 0 ) = l i m 𝑑 β†’ + ∞ 𝑒 ( 𝑑 ) , l i m 𝑑 β†’ 0 + | | 𝑒 𝑀 ( 𝑑 ) ξ…ž | | 𝑝 ( 𝑑 ) βˆ’ 2 𝑒 ξ…ž ( 𝑑 ) = l i m 𝑑 β†’ + ∞ | | 𝑒 𝑀 ( 𝑑 ) ξ…ž | | 𝑝 ( 𝑑 ) βˆ’ 2 𝑒 ξ…ž ( 𝑑 ) , ( 1 . 2 ) where 𝑝 ∈ 𝐢 ( [ 0 , + ∞ ) , ℝ ) , 𝑝 ( 𝑑 ) > 1 , l i m 𝑑 β†’ + ∞ 𝑝 ( 𝑑 ) exists and l i m 𝑑 β†’ + ∞ 𝑝 ( 𝑑 ) > 1 , βˆ’ Ξ” 𝑝 ( 𝑑 ) 𝑒 = βˆ’ ( 𝑀 ( 𝑑 ) | 𝑒 ξ…ž | 𝑝 ( 𝑑 ) βˆ’ 2 𝑒 ξ…ž ) ξ…ž is called the weighted 𝑝 ( 𝑑 ) -Laplacian; 𝑀 ∈ 𝐢 ( [ 0 , + ∞ ) , ℝ ) satisfies 0 < 𝑀 ( 𝑑 ) , f o r a l l 𝑑 ∈ ( 0 , + ∞ ) , and ( 𝑀 ( 𝑑 ) ) βˆ’ 1 / ( 𝑝 ( 𝑑 ) βˆ’ 1 ) ∈ 𝐿 1 ( 0 , + ∞ ) ; the equivalent l i m π‘Ÿ β†’ 0 + 𝑀 ( π‘Ÿ ) | 𝑒 ξ…ž | 𝑝 ( π‘Ÿ ) βˆ’ 2 𝑒 ξ…ž ( π‘Ÿ ) = l i m π‘Ÿ β†’ + ∞ 𝑀 ( π‘Ÿ ) | 𝑒 ξ…ž | 𝑝 ( π‘Ÿ ) βˆ’ 2 𝑒 ξ…ž ( π‘Ÿ ) means that l i m π‘Ÿ β†’ 0 + 𝑀 ( π‘Ÿ ) | 𝑒 ξ…ž | 𝑝 ( π‘Ÿ ) βˆ’ 2 𝑒 ξ…ž ( π‘Ÿ ) and l i m π‘Ÿ β†’ + ∞ 𝑀 ( π‘Ÿ ) | 𝑒 ξ…ž | 𝑝 ( π‘Ÿ ) βˆ’ 2 𝑒 ξ…ž ( π‘Ÿ ) both exist and equal; 𝛿 is a positive parameter.

The study of differential equations and variational problems with variable exponent growth conditions is a new and interesting topic. Many results have been obtained on these kinds of problems, for example, [1–15]. We refer to [2, 16, 17], the applied background on these problems. If 𝑀 ( 𝑑 ) ≑ 1 and 𝑝 ( 𝑑 ) ≑ 𝑝 (a constant), βˆ’ Ξ” 𝑝 ( 𝑑 ) is the well-known 𝑝 -Laplacian. If 𝑝 ( 𝑑 ) is a general function, βˆ’ Ξ” 𝑝 ( 𝑑 ) represents a nonhomogeneity and possesses more nonlinearity, and thus βˆ’ Ξ” 𝑝 ( 𝑑 ) is more complicated than βˆ’ Ξ” 𝑝 . For example, We have the following.

(1)If Ξ© βŠ‚ ℝ 𝑛 is a bounded domain, the Rayleigh quotient πœ† 𝑝 ( π‘₯ ) = i n f 𝑒 ∈ π‘Š 0 1 , 𝑝 ( π‘₯ ) ( Ξ© ) ⧡ { 0 } ∫ Ξ© | | | | ( 1 / 𝑝 ( π‘₯ ) ) βˆ‡ 𝑒 𝑝 ( π‘₯ ) 𝑑 π‘₯ ∫ Ξ© ( 1 / 𝑝 ( π‘₯ ) ) | 𝑒 | 𝑝 ( π‘₯ ) 𝑑 π‘₯ ( 1 . 3 ) is zero in general, and only under some special conditions πœ† 𝑝 ( π‘₯ ) > 0 (see [6]), but the fact that πœ† 𝑝 > 0 is very important in the study of 𝑝 -Laplacian problems;(2)If 𝑀 ( 𝑑 ) ≑ 1 and 𝑝 ( 𝑑 ) ≑ 𝑝 (a constant) and βˆ’ Ξ” 𝑝 𝑒 > 0 , then 𝑒 is concave; this property is used extensively in the study of one dimensional 𝑝 -Laplacian problems, but it is invalid for βˆ’ Ξ” 𝑝 ( 𝑑 ) . It is another difference on βˆ’ Ξ” 𝑝 and βˆ’ Ξ” 𝑝 ( 𝑑 ) .(3)On the existence of solutions of the following typical βˆ’ Ξ” 𝑝 ( 𝑑 ) problem; βˆ’ ξ‚€ | | 𝑒 ξ…ž | | 𝑝 ( π‘₯ ) βˆ’ 2 𝑒 ξ…ž  ξ…ž = | 𝑒 | π‘ž ( π‘₯ ) βˆ’ 2 𝑒 + 𝐢 , π‘₯ ∈ Ξ© βŠ‚ ℝ 𝑁 , 𝑒 = 0 o n πœ• Ξ© , ( 1 . 4 ) because of the nonhomogeneity of βˆ’ Ξ” 𝑝 ( π‘₯ ) , and if 1 ≀ m a x π‘₯ ∈ Ξ© π‘ž ( π‘₯ ) < m i n π‘₯ ∈ Ξ© 𝑝 ( π‘₯ ) , then the corresponding functional is coercive, if m a x π‘₯ ∈ Ξ© 𝑝 ( π‘₯ ) < m i n π‘₯ ∈ Ξ© π‘ž ( π‘₯ ) , then the corresponding functional can satisfy Palais-Smale condition, (see [4, 7]). If m i n π‘₯ ∈ Ξ© 𝑝 ( π‘₯ ) ≀ π‘ž ( π‘₯ ) ≀ m a x π‘₯ ∈ Ξ© 𝑝 ( π‘₯ ) , there are more difficulties to testify that the corresponding functional is coercive or satisfying Palais-Smale conditions, and the results on this case are rare.

There are many results on the existence of solutions for 𝑝 -Laplacian equation with multi-point boundary value conditions (see [18–21]). On the existence of solutions for 𝑝 ( π‘₯ ) -Laplacian systems boundary value problems, we refer to [5, 7, 10–15]. But results on the existence and asymptotic behavior of solutions for weighted 𝑝 ( 𝑑 ) -Laplacian systems with multi-point boundary value conditions are rare. In this paper, when 𝑝 ( 𝑑 ) is a general function, we investigate the existence and asymptotic behavior of solutions for weighted 𝑝 ( 𝑑 ) -Laplacian systems with multi-point boundary value conditions. Moreover, the case of m i n 𝑑 ∈ [ 0 , 1 ] 𝑝 ( 𝑑 ) ≀ π‘ž ( 𝑑 ) ≀ m a x 𝑑 ∈ [ 0 , 1 ] 𝑝 ( 𝑑 ) has been discussed.

Let 𝑁 β‰₯ 1 and 𝐼 = [ 0 , + ∞ ) ; the function 𝑓 = ( 𝑓 1 , … , 𝑓 𝑁 ) ∢ 𝐼 Γ— ℝ 𝑁 Γ— ℝ 𝑁 β†’ ℝ 𝑁 is assumed to be Caratheodory, by this we mean that

(i)for almost every 𝑑 ∈ 𝐼 , the function 𝑓 ( 𝑑 , β‹… , β‹… ) is continuous;(ii)for each ( π‘₯ , 𝑦 ) ∈ ℝ 𝑁 Γ— ℝ 𝑁 , the function 𝑓 ( β‹… , π‘₯ , 𝑦 ) is measurable on 𝐼 ;(iii)for each 𝑅 > 0 , there is a 𝛽 𝑅 ∈ 𝐿 1 ( 𝐼 , ℝ ) such that, for almost every 𝑑 ∈ 𝐼 and every ( π‘₯ , 𝑦 ) ∈ ℝ 𝑁 Γ— ℝ 𝑁 with | π‘₯ | ≀ 𝑅 , | 𝑦 | ≀ 𝑅 , one has | | | | 𝑓 ( 𝑑 , π‘₯ , 𝑦 ) ≀ 𝛽 𝑅 ( 𝑑 ) . ( 1 . 5 )

Throughout the paper, we denote | | 𝑒 𝑀 ( 0 ) ξ…ž | | 𝑝 ( 0 ) βˆ’ 2 𝑒 ξ…ž ( 0 ) = l i m 𝑑 β†’ 0 + | | 𝑒 𝑀 ( 𝑑 ) ξ…ž | | 𝑝 ( 𝑑 ) βˆ’ 2 𝑒 ξ…ž | | 𝑒 ( 𝑑 ) , 𝑀 ( + ∞ ) ξ…ž | | 𝑝 ( + ∞ ) βˆ’ 2 𝑒 ξ…ž ( + ∞ ) = l i m 𝑑 β†’ + ∞ | | 𝑒 𝑀 ( 𝑑 ) ξ…ž | | 𝑝 ( 𝑑 ) βˆ’ 2 𝑒 ξ…ž ( 𝑑 ) . ( 1 . 6 )

The inner product in ℝ 𝑁 will be denoted by ⟨ β‹… , β‹… ⟩ , | β‹… | will denote the absolute value and the Euclidean norm on ℝ 𝑁 . Let 𝐴 𝐢 ( 0 , + ∞ ) denote the space of absolutely continuous functions on the interval ( 0 , + ∞ ) . For 𝑁 β‰₯ 1 , we set 𝐢 = 𝐢 ( 𝐼 , ℝ 𝑁 ) , 𝐢 1 = { 𝑒 ∈ 𝐢 ∣ 𝑒 ξ…ž ∈ 𝐢 ( ( 0 , + ∞ ) , ℝ 𝑁 ) , l i m 𝑑 β†’ 0 + 𝑀 ( 𝑑 ) 1 / ( 𝑝 ( 𝑑 ) βˆ’ 1 ) 𝑒 ξ…ž ( 𝑑 ) e x i s t s } . For any 𝑒 ( 𝑑 ) = ( 𝑒 1 ( 𝑑 ) , … , 𝑒 𝑁 ( 𝑑 ) ) , we denote | 𝑒 𝑖 | 0 = s u p 𝑑 ∈ ( 0 , + ∞ ) | 𝑒 𝑖 ( 𝑑 ) | ,   β€– 𝑒 β€– 0 βˆ‘ = ( 𝑁 𝑖 = 1 | 𝑒 𝑖 | 2 0 ) 1 / 2 and β€– 𝑒 β€– 1 = β€– 𝑒 β€– 0 + β€– ( 𝑀 ( 𝑑 ) ) 1 / ( 𝑝 ( 𝑑 ) βˆ’ 1 ) 𝑒 ξ…ž β€– 0 . Spaces 𝐢 and 𝐢 1 will be equipped with the norm β€– β‹… β€– 0 and β€– β‹… β€– 1 , respectively. Then ( 𝐢 , β€– β‹… β€– 0 ) and ( 𝐢 1 , β€– β‹… β€– 1 ) are Banach spaces. Denote 𝐿 1 = 𝐿 1 ( 𝐼 , ℝ 𝑁 ) , the norm β€– 𝑒 β€– 𝐿 1 βˆ‘ = [ 𝑁 𝑖 = 1 ( ∫ ∞ 0 | 𝑒 𝑖 | 𝑑 𝑑 ) 2 ] 1 / 2 .

We say a function 𝑒 ∢ 𝐼 β†’ ℝ 𝑁 is a solution of (1.1) if 𝑒 ∈ 𝐢 1 with 𝑀 ( 𝑑 ) | 𝑒 ξ…ž | 𝑝 ( 𝑑 ) βˆ’ 2 𝑒 ξ…ž ( 𝑑 ) absolutely continuous on ( 0 , + ∞ ), which satisfies (1.1) almost every on 𝐼 .

In this paper, we always use 𝐢 𝑖 to denote positive constants, if it cannot lead to confusion. Denote 𝑧 βˆ’ = m i n 𝑑 ∈ 𝐼 𝑧 ( 𝑑 ) , 𝑧 + = m a x 𝑑 ∈ 𝐼 𝑧 ( 𝑑 ) , f o r a n y 𝑧 ∈ 𝐢 ( 𝐼 , ℝ ) . ( 1 . 7 )

We say 𝑓 satisfies sub-( 𝑝 βˆ’ βˆ’ 1 ) growth condition, if 𝑓 satisfies l i m | 𝑒 | + | 𝑣 | β†’ + ∞ ξ‚΅ 𝑓 ( 𝑑 , 𝑒 , 𝑣 ) ( | 𝑒 | + | 𝑣 | ) π‘ž ( 𝑑 ) βˆ’ 1 ξ‚Ά = 0 , f o r 𝑑 ∈ 𝐼 u n i f o r m l y , ( 1 . 8 ) where π‘ž ( 𝑑 ) ∈ 𝐢 ( 𝐼 , ℝ ) , and 1 < π‘ž βˆ’ ≀ π‘ž + < 𝑝 βˆ’ . We say 𝑓 satisfies general growth condition, if we don't know whether 𝑓 satisfies sub-( 𝑝 βˆ’ βˆ’ 1 ) growth condition or not.

We will discuss the existence of solutions of (1.1)-(1.2) in the following two cases

(i) 𝑓 satisfies sub-( 𝑝 βˆ’ βˆ’ 1 ) growth condition;(ii) 𝑓 satisfies general growth condition.

This paper is divided into four sections. In the second section, we will do some preparation. In the third section, we will discuss the existence and asymptotic behavior of solutions of (1.1)-(1.2), when 𝑓 satisfies sub-( 𝑝 βˆ’ βˆ’ 1 ) growth condition. Finally, in the fourth section, we will discuss the existence and asymptotic behavior of solutions of (1.1)-(1.2), when 𝑓 satisfies general growth condition.

2. Preliminary

For any ( 𝑑 , π‘₯ ) ∈ 𝐼 Γ— ℝ 𝑁 , denote πœ‘ ( 𝑑 , π‘₯ ) = | π‘₯ | 𝑝 ( 𝑑 ) βˆ’ 2 π‘₯ . Obviously, πœ‘ has the following properties.

Lemma 2.1 (see [4]). πœ‘ is a continuous function and satisfies (i)For any 𝑑 ∈ [ 0 , + ∞ ) , πœ‘ ( 𝑑 , β‹… ) is strictly monotone, that is,  πœ‘ ξ€· 𝑑 , π‘₯ 1 ξ€Έ ξ€· βˆ’ πœ‘ 𝑑 , π‘₯ 2 ξ€Έ , π‘₯ 1 βˆ’ π‘₯ 2  > 0 , f o r a n y π‘₯ 1 , π‘₯ 2 ∈ ℝ 𝑁 , π‘₯ 1 β‰  π‘₯ 2 . ( 2 . 1 ) (ii)There exists a function 𝛽 ∢ [ 0 , + ∞ ) β†’ [ 0 , + ∞ ) , 𝛽 ( 𝑠 ) β†’ + ∞ as 𝑠 β†’ + ∞ , such that ⟨ πœ‘ ( 𝑑 , π‘₯ ) , π‘₯ ⟩ β‰₯ 𝛽 ( | π‘₯ | ) | π‘₯ | , βˆ€ π‘₯ ∈ ℝ 𝑁 . ( 2 . 2 )

It is well known that πœ‘ ( 𝑑 , β‹… ) is a homeomorphism from ℝ 𝑁 to ℝ 𝑁 for any fixed 𝑑 ∈ [ 0 , + ∞ ) . For any 𝑑 ∈ 𝐼 , denote by πœ‘ βˆ’ 1 ( 𝑑 , β‹… ) the inverse operator of πœ‘ ( 𝑑 , β‹… ) , then πœ‘ βˆ’ 1 ( 𝑑 , π‘₯ ) = | π‘₯ | ( 2 βˆ’ 𝑝 ( 𝑑 ) ) / ( 𝑝 ( 𝑑 ) βˆ’ 1 ) π‘₯ , f o r π‘₯ ∈ ℝ 𝑁 ⧡ { 0 } , πœ‘ βˆ’ 1 ( 𝑑 , 0 ) = 0 . ( 2 . 3 ) It is clear that πœ‘ βˆ’ 1 ( 𝑑 , β‹… ) is continuous and sends bounded sets into bounded sets. Let us now consider the following problem with boundary value condition (1.2): ξ€· ξ€· 𝑀 ( 𝑑 ) πœ‘ 𝑑 , 𝑒 ξ…ž ( 𝑑 ) ξ€Έ ξ€Έ ξ…ž = 𝑔 ( 𝑑 ) , 𝑑 ∈ ( 0 , + ∞ ) , ( 2 . 4 ) where 𝑔 ∈ 𝐿 1 , and satisfies ∫ 0 + ∞ 𝑔 ( 𝑑 ) 𝑑 𝑑 = 0 . If 𝑒 is a solution of (2.4) with (1.2), by integrating (2.4) from 0 to 𝑑 , we find that ξ€· 𝑀 ( 𝑑 ) πœ‘ 𝑑 , 𝑒 ξ…ž ( ξ€Έ ξ€· 𝑑 ) = 𝑀 ( 0 ) πœ‘ 0 , 𝑒 ξ…ž ( ξ€Έ + ξ€œ 0 ) 𝑑 0 𝑔 ( 𝑠 ) 𝑑 𝑠 . ( 2 . 5 ) Denote π‘Ž = 𝑀 ( 0 ) πœ‘ ( 0 , 𝑒 ξ…ž ( 0 ) ) . It is easy to see that π‘Ž is dependent on 𝑔 ( 𝑑 ) . Define operator 𝐹 ∢ 𝐿 1 β†’ 𝐢 as ξ€œ 𝐹 ( 𝑔 ) ( 𝑑 ) = 𝑑 0 𝑔 ( 𝑠 ) 𝑑 𝑠 , 𝑑 ∈ 𝐼 , 𝑔 ∈ 𝐿 1 . ( 2 . 6 ) By solving for 𝑒 ξ…ž in (2.5) and integrating, we find that ξ€½ πœ‘ 𝑒 ( 𝑑 ) = 𝑒 ( 0 ) + 𝐹 βˆ’ 1 ξ€Ί 𝑑 , ( 𝑀 ( 𝑑 ) ) βˆ’ 1 ( π‘Ž + 𝐹 ( 𝑔 ) ) ξ€» ξ€Ύ ( 𝑑 ) , 𝑑 ∈ 𝐼 . ( 2 . 7 ) The boundary condition (1.2) implies that ξ€œ 0 + ∞ πœ‘ βˆ’ 1 ξ€½ 𝑑 , ( 𝑀 ( 𝑑 ) ) βˆ’ 1 [ ] ξ€Ύ π‘Ž + 𝐹 ( 𝑔 ) ( 𝑑 ) 𝑑 𝑑 = 0 . ( 2 . 8 ) For fixed β„Ž ∈ 𝐢 , we denote Ξ› β„Ž ξ€œ ( π‘Ž ) = 0 + ∞ πœ‘ βˆ’ 1 ξ€½ 𝑑 , ( 𝑀 ( 𝑑 ) ) βˆ’ 1 [ ] ξ€Ύ π‘Ž + β„Ž ( 𝑑 ) 𝑑 𝑑 . ( 2 . 9 ) Throughout the paper, we denote ∫ 𝐸 = 0 + ∞ ( 𝑀 ( 𝑑 ) ) βˆ’ 1 / ( 𝑝 ( 𝑑 ) βˆ’ 1 ) 𝑑 𝑑 .

Lemma 2.2. The function Ξ› β„Ž ( β‹… ) has the following properties. (i)For any fixed β„Ž ∈ 𝐢 , the equation Ξ› β„Ž ( π‘Ž ) = 0 ( 2 . 1 0 ) has a unique solution Μƒ π‘Ž ( β„Ž ) ∈ ℝ 𝑁 .(ii)The function Μƒ π‘Ž ∢ 𝐢 β†’ ℝ 𝑁 , defined in ( i ) , is continuous and sends bounded sets to bounded sets. Moreover | | | | Μƒ π‘Ž ( β„Ž ) ≀ 3 𝑁 β€– β„Ž β€– 0 . ( 2 . 1 1 )

Proof. (i) From Lemma 2.1, it is immediate that  Ξ› β„Ž ξ€· π‘Ž 1 ξ€Έ βˆ’ Ξ› β„Ž ξ€· π‘Ž 2 ξ€Έ , π‘Ž 1 βˆ’ π‘Ž 2  > 0 , f o r π‘Ž 1 β‰  π‘Ž 2 , ( 2 . 1 2 ) and hence, if (2.10) has a solution, then it is unique.
Let 𝑑 0 = 3 𝑁 β€– β„Ž β€– 0 . If | π‘Ž | > 𝑑 0 , since ( 𝑀 ( 𝑑 ) ) βˆ’ 1 / ( 𝑝 ( 𝑑 ) βˆ’ 1 ) ∈ 𝐿 1 ( 0 , + ∞ ) and β„Ž ∈ 𝐢 , it is easy to see that there exists an 𝑖 ∈ { 1 , … , 𝑁 } such that the 𝑖 th component π‘Ž 𝑖 of π‘Ž satisfies | π‘Ž 𝑖 | > 3 β€– β„Ž β€– 0 . Thus ( π‘Ž 𝑖 + β„Ž 𝑖 ( 𝑑 ) ) keeps sign on 𝐼 and
| | π‘Ž 𝑖 + β„Ž 𝑖 | | β‰₯ | | π‘Ž ( 𝑑 ) 𝑖 | | βˆ’ β€– β„Ž β€– 0 > 2 β€– β„Ž β€– 0 f o r a n y 𝑑 ∈ 𝐼 , ( 2 . 1 3 ) then | | π‘Ž 𝑖 + β„Ž 𝑖 | | ( 𝑑 ) 1 / ( 𝑝 ( 𝑑 ) βˆ’ 1 ) > ξ€Ί 2 β€– β„Ž β€– 0 ξ€» 1 / ( 𝑝 ( πœ‰ ) βˆ’ 1 ) , w h e r e πœ‰ ∈ 𝐼 , f o r a n y 𝑑 ∈ 𝐼 . ( 2 . 1 4 )
Thus the 𝑖 th component Ξ› 𝑖 β„Ž ( π‘Ž ) of Ξ› β„Ž ( π‘Ž ) is nonzero and keeps sign, and then we have
ξ€œ 0 + ∞ πœ‘ βˆ’ 1 ξ€½ 𝑑 , ( 𝑀 ( 𝑑 ) ) βˆ’ 1 [ ] ξ€Ύ π‘Ž + β„Ž ( 𝑑 ) 𝑑 𝑑 β‰  0 . ( 2 . 1 5 )
Let us consider the equation
πœ† Ξ› β„Ž [ ] . ( π‘Ž ) + ( 1 βˆ’ πœ† ) π‘Ž = 0 , πœ† ∈ 0 , 1 ( 2 . 1 6 )
It is easy to see that all the solutions of (2.16) belong to 𝑏 ( 𝑑 0 + 1 ) = { π‘₯ ∈ ℝ 𝑁 ∣ | π‘₯ | < 𝑑 0 + 1 } . So, we have
𝑑 𝐡 ξ€Ί Ξ› β„Ž ξ€· 𝑑 ( π‘Ž ) , 𝑏 0 ξ€Έ ξ€» + 1 , 0 = 𝑑 𝐡 ξ€Ί ξ€· 𝑑 𝐼 , 𝑏 0 ξ€Έ ξ€» + 1 , 0 β‰  0 , ( 2 . 1 7 ) and it means the existence of solutions of Ξ› β„Ž ( π‘Ž ) = 0 .
In this way, we define a function Μƒ π‘Ž ( β„Ž ) ∢ 𝐢 [ 0 , + ∞ ) β†’ ℝ 𝑁 , which satisfies Ξ› β„Ž ( Μƒ π‘Ž ( β„Ž ) ) = 0 . ( 2 . 1 8 )
(ii) By the proof of (i), we also obtain Μƒ π‘Ž sends bounded sets to bounded sets, and | | | | Μƒ π‘Ž ( β„Ž ) ≀ 3 𝑁 β€– β„Ž β€– 0 . ( 2 . 1 9 )
It only remains to prove the continuity of Μƒ π‘Ž . Let { 𝑒 𝑛 } be a convergent sequence in 𝐢 and 𝑒 𝑛 β†’ 𝑒 as 𝑛 β†’ + ∞ . Since { Μƒ π‘Ž ( 𝑒 𝑛 ) } is a bounded sequence, then it contains a convergent subsequence { Μƒ π‘Ž ( 𝑒 𝑛 𝑗 ) } . Let Μƒ π‘Ž ( 𝑒 𝑛 𝑗 ) β†’ π‘Ž 0 as 𝑗 β†’ + ∞ . Since Ξ› 𝑒 𝑛 𝑗 ( Μƒ π‘Ž ( 𝑒 𝑛 𝑗 ) ) = 0 , letting 𝑗 β†’ + ∞ , we have Ξ› 𝑒 ( π‘Ž 0 ) = 0 . From (i), we get π‘Ž 0 = Μƒ π‘Ž ( 𝑒 ) , and it means that Μƒ π‘Ž is continuous. This completes the proof.

Now, we define the operator π‘Ž ∢ 𝐿 1 β†’ ℝ 𝑁 as π‘Ž ( 𝑒 ) = Μƒ π‘Ž ( 𝐹 ( 𝑒 ) ) . ( 2 . 2 0 )

It is clear that π‘Ž ( β‹… ) is continuous and sends bounded sets of 𝐿 1 to bounded sets of ℝ 𝑁 , and hence it is a compact continuous mapping.

If 𝑒 is a solution of (2.4) with (1.2), then ξ€½ πœ‘ 𝑒 ( 𝑑 ) = 𝑒 ( 0 ) + 𝐹 βˆ’ 1 ξ€Ί 𝑑 , ( 𝑀 ( 𝑑 ) ) βˆ’ 1 [ ( π‘Ž ( 𝑔 ) + 𝐹 ( 𝑔 ) ( 𝑑 ) ) ξ€» ξ€Ύ ( 𝑑 ) , βˆ€ 𝑑 ∈ 0 , + ∞ ) . ( 2 . 2 1 ) Let us define 𝑃 ∢ 𝐢 1 ⟢ 𝐢 1 , 𝑒 ⟼ 𝑒 ( 0 ) ; 𝑄 ∢ 𝐿 1 ⟢ ℝ 𝑁 ξ€œ , β„Ž ⟼ 0 + ∞ 𝑄 β„Ž ( π‘Ÿ ) 𝑑 π‘Ÿ ; βˆ— ∢ 𝐿 1 ⟢ 𝐿 1 ξ€œ , β„Ž ⟼ 𝜏 ( 𝑑 ) 0 + ∞ β„Ž ( π‘Ÿ ) 𝑑 π‘Ÿ ; ( 2 . 2 2 ) where 𝜏 ∈ ( [ 0 , ∞ ) , 𝑅 ) and satisfies 0 < 𝜏 ( 𝑑 ) < 1 , ∫ 𝑑 ∈ 𝐼 , 0 + ∞ 𝜏 ( 𝑑 ) 𝑑 𝑑 = 1 , and we denote 𝐾 1 ∢ 𝐿 1 β†’ 𝐢 1 as 𝐾 1 ξ€· 𝐾 ( β„Ž ) ( 𝑑 ) ∢ = 1 ξ€Έ ξ€½ πœ‘ ∘ β„Ž ( 𝑑 ) = 𝐹 βˆ’ 1 ξ€Ί 𝑑 , ( 𝑀 ( 𝑑 ) ) βˆ’ 1 ξ€· π‘Ž ξ€· ξ€· 𝐼 βˆ’ 𝑄 βˆ— ξ€Έ β„Ž ξ€Έ + 𝐹 ξ€· ξ€· 𝐼 βˆ’ 𝑄 βˆ— ξ€Έ β„Ž [ ξ€Έ ξ€Έ ξ€» ξ€Ύ ( 𝑑 ) , βˆ€ 𝑑 ∈ 0 , + ∞ ) . ( 2 . 2 3 )

Lemma 2.3. The operator 𝐾 1 is continuous and sends equi-integrable sets in 𝐿 1 to relatively compact sets in 𝐢 1 .

Proof. It is easy to check that 𝐾 1 ( β„Ž ) ( β‹… ) ∈ 𝐢 1 , f o r a l l β„Ž ∈ 𝐿 1 . Since ( 𝑀 ( 𝑑 ) ) βˆ’ 1 / ( 𝑝 ( 𝑑 ) βˆ’ 1 ) ∈ 𝐿 1 and 𝐾 1 ( β„Ž ) ξ…ž ( 𝑑 ) = πœ‘ βˆ’ 1 ξ€Ί 𝑑 , ( 𝑀 ( 𝑑 ) ) βˆ’ 1 ξ€· π‘Ž ξ€· ξ€· 𝐼 βˆ’ 𝑄 βˆ— ξ€Έ β„Ž ξ€Έ + 𝐹 ξ€· ξ€· 𝐼 βˆ’ 𝑄 βˆ— ξ€Έ β„Ž [ ξ€Έ ξ€Έ ξ€» , βˆ€ 𝑑 ∈ 0 , + ∞ ) , ( 2 . 2 4 ) it is easy to check that 𝐾 1 is a continuous operator from 𝐿 1 to 𝐢 1 .
Let now π‘ˆ be an equi-integrable set in 𝐿 1 , then there exists 𝜌 ∈ 𝐿 1 , such that
| | | | 𝑒 ( 𝑑 ) ≀ 𝜌 ( 𝑑 ) a . e . i n 𝐼 , f o r a n y 𝑒 ∈ 𝐿 1 . ( 2 . 2 5 )
We want to show that 𝐾 1 ( π‘ˆ ) βŠ‚ 𝐢 1 is a compact set.
Let { 𝑒 𝑛 } be a sequence in 𝐾 1 ( π‘ˆ ) , then there exists a sequence { β„Ž 𝑛 } ∈ π‘ˆ such that 𝑒 𝑛 = 𝐾 1 ( β„Ž 𝑛 ) . For any 𝑑 1 , 𝑑 2 ∈ 𝐼 , we have that
| | 𝐹 ξ€· ξ€· 𝐼 βˆ’ 𝑄 βˆ— ξ€Έ β„Ž 𝑛 𝑑 ξ€Έ ξ€· 1 ξ€Έ βˆ’ 𝐹 ξ€· ξ€· 𝐼 βˆ’ 𝑄 βˆ— ξ€Έ β„Ž 𝑛 𝑑 ξ€Έ ξ€· 2 ξ€Έ | | ≀ | | | | ξ€œ 𝑑 2 𝑑 1 𝜌 | | | | + | | | | ξ€œ ( 𝑑 ) 𝑑 𝑑 𝑑 2 𝑑 1 𝜏 | | | | ( 𝑑 ) 𝑑 𝑑 𝑄 𝜌 . ( 2 . 2 6 )
Hence the sequence { 𝐹 ( ( 𝐼 βˆ’ 𝑄 βˆ— ) β„Ž 𝑛 ) } is equicontinuous.
From the definition of 𝑄 βˆ— , we have 𝐹 ( ( 𝐼 βˆ’ 𝑄 βˆ— ) β„Ž 𝑛 ) ( + ∞ ) = 0 , 𝑛 = 1 , 2 , … . Thus
| | 𝐹 ξ€· ξ€· 𝐼 βˆ’ 𝑄 βˆ— ξ€Έ β„Ž 𝑛 ξ€Έ | | = | | 𝐹 ( 𝑑 ) ξ€· ξ€· 𝐼 βˆ’ 𝑄 βˆ— ξ€Έ β„Ž 𝑛 ξ€Έ ( 𝑑 ) βˆ’ 𝐹 ξ€· ξ€· 𝐼 βˆ’ 𝑄 βˆ— ξ€Έ β„Ž 𝑛 ξ€Έ | | = | | | | ξ€œ ( + ∞ ) 𝑑 + ∞ ξ€· ξ€· 𝐼 βˆ’ 𝑄 βˆ— ξ€Έ β„Ž 𝑛 ξ€Έ ( | | | | ≀ | | | | ξ€œ 𝑑 ) 𝑑 𝑑 𝑑 + ∞ β„Ž 𝑛 | | | | + ξ€œ ( 𝑑 ) 𝑑 𝑑 𝑑 + ∞ 𝜏 | | ( 𝑑 ) 𝑑 𝑑 β‹… 𝑄 β„Ž 𝑛 | | ≀ ξ€œ 𝑑 + ∞ ξ€œ 𝜌 ( 𝑑 ) 𝑑 𝑑 + 𝑑 + ∞ 𝜏 ( 𝑑 ) 𝑑 𝑑 β‹… 𝑄 ( 𝜌 ) ⟢ 0 , a s 𝑑 ⟢ + ∞ . ( 2 . 2 7 )
Thus { 𝐹 ( ( 𝐼 βˆ’ 𝑄 βˆ— ) β„Ž 𝑛 ) } is uniformly bounded.
By Ascoli-Arzela theorem, there exists a subsequence of { 𝐹 ( ( 𝐼 βˆ’ 𝑄 βˆ— ) β„Ž 𝑛 ) } (which we rename the same) being convergent in 𝐢 . According to the bounded continuous of the operator π‘Ž , we can choose a subsequence of { π‘Ž ( ( 𝐼 βˆ’ 𝑄 βˆ— ) β„Ž 𝑛 ) + 𝐹 ( ( 𝐼 βˆ’ 𝑄 βˆ— ) β„Ž 𝑛 ) } (which we still denote { π‘Ž ( ( 𝐼 βˆ’ 𝑄 βˆ— ) β„Ž 𝑛 ) + 𝐹 ( ( 𝐼 βˆ’ 𝑄 βˆ— ) β„Ž 𝑛 ) } is convergent in 𝐢 , then 𝑀 ( 𝑑 ) πœ‘ ( 𝑑 , 𝐾 1 ( β„Ž 𝑛 ) ξ…ž ( 𝑑 ) ) = π‘Ž ( ( 𝐼 βˆ’ 𝑄 βˆ— ) β„Ž 𝑛 ) + 𝐹 ( ( 𝐼 βˆ’ 𝑄 βˆ— ) β„Ž 𝑛 ) is convergent in 𝐢 .
Since
𝐾 1 ξ€· β„Ž 𝑛 ξ€Έ ξ€½ πœ‘ ( 𝑑 ) = 𝐹 βˆ’ 1 ξ€Ί π‘Ÿ , ( 𝑀 ( π‘Ÿ ) ) βˆ’ 1 ξ€· π‘Ž ξ€· ξ€· 𝐼 βˆ’ 𝑄 βˆ— ξ€Έ β„Ž 𝑛 ξ€Έ + 𝐹 ξ€· ξ€· 𝐼 βˆ’ 𝑄 βˆ— ξ€Έ β„Ž 𝑛 [ ξ€Έ ξ€Έ ξ€» ξ€Ύ ( 𝑑 ) , βˆ€ 𝑑 ∈ 0 , + ∞ ) , ( 2 . 2 8 ) from the continuity of πœ‘ βˆ’ 1 and the integrability of 𝑀 ( 𝑑 ) βˆ’ 1 / ( 𝑝 ( 𝑑 ) βˆ’ 1 ) in 𝐿 1 , we can see that 𝐾 1 ( β„Ž 𝑛 ) is convergent in 𝐢 . Thus that { 𝑒 𝑛 } is convergent in 𝐢 1 .
This completes the proof.

We denote by 𝑁 𝑓 ( 𝑒 ) ∢ [ 0 , + ∞ ) Γ— 𝐢 1 β†’ 𝐿 1 the Nemytski operator associated to 𝑓 defined by 𝑁 𝑓 ξ€· ( 𝑒 ) ( 𝑑 ) = 𝑓 𝑑 , 𝑒 ( 𝑑 ) , ( 𝑀 ( 𝑑 ) ) 1 / ( 𝑝 ( 𝑑 ) βˆ’ 1 ) 𝑒 ξ…ž ξ€Έ , ( 𝑑 ) a . e . o n 𝐼 . ( 2 . 2 9 )

Lemma 2.4. 𝑒 is a solution of (1.1)-(1.2) if and only if 𝑒 is a solution of the following abstract equation: 𝑒 = 𝑃 𝑒 + 𝑄 𝛿 𝑁 𝑓 ( 𝑒 ) + 𝐾 1 ξ€· 𝛿 𝑁 𝑓 ξ€Έ . ( 𝑒 ) ( 2 . 3 0 )

Proof. If 𝑒 is a solution of (1.1)-(1.2), by integrating (1.1) from 0 to 𝑑 , we find that 𝑀 ξ€· ( 𝑑 ) πœ‘ 𝑑 , 𝑒 ξ…ž ξ€Έ ξ€· ( 𝑑 ) = π‘Ž 𝛿 𝑁 𝑓 ξ€Έ ξ€· ( 𝑒 ) + 𝐹 𝛿 𝑁 𝑓 ξ€Έ ( 𝑒 ) ( 𝑑 ) , βˆ€ 𝑑 ∈ ( 0 , + ∞ ) . ( 2 . 3 1 )
From (2.31), we have
ξ€½ πœ‘ 𝑒 ( 𝑑 ) = 𝑒 ( 0 ) + 𝐹 βˆ’ 1 ξ€Ί 𝑑 , ( 𝑀 ( 𝑑 ) ) βˆ’ 1 ξ€· π‘Ž ξ€· 𝛿 𝑁 𝑓 ξ€Έ ξ€· + 𝐹 𝛿 𝑁 𝑓 [ ( 𝑒 ) ξ€Έ ξ€Έ ξ€» ξ€Ύ ( 𝑑 ) , βˆ€ 𝑑 ∈ 0 , + ∞ ) . ( 2 . 3 2 )
From 𝑀 ( 0 ) | 𝑒 ξ…ž | 𝑝 ( 0 ) βˆ’ 2 𝑒 ξ…ž ( 0 ) = 𝑀 ( + ∞ ) | 𝑒 ξ…ž | 𝑝 ( + ∞ ) βˆ’ 2 𝑒 ξ…ž ( + ∞ ) , we have
𝑄 𝛿 𝑁 𝑓 𝑄 ( 𝑒 ) = 0 , βˆ— 𝛿 𝑁 𝑓 ( 𝑒 ) = 0 . ( 2 . 3 3 )
So we have
𝑒 = 𝑃 𝑒 + 𝑄 𝛿 𝑁 𝑓 ( 𝑒 ) + 𝐾 1 ξ€· 𝛿 𝑁 𝑓 ξ€Έ . ( 𝑒 ) ( 2 . 3 4 )
Conversely, if 𝑒 is a solution of (2.30), then
𝑒 ( 0 ) = 𝑃 𝑒 + 𝑄 𝛿 𝑁 𝑓 ( 𝑒 ) + 𝐾 1 ξ€· 𝛿 𝑁 𝑓 ξ€Έ ( 𝑒 ) ( 0 ) = 𝑒 ( 0 ) + 𝑄 𝛿 𝑁 𝑓 ( 𝑒 ) . ( 2 . 3 5 )
Thus 𝑄 𝛿 𝑁 𝑓 ( 𝑒 ) = 0 and 𝑄 βˆ— 𝛿 𝑁 𝑓 ( 𝑒 ) = 0 . By the definition of the mapping π‘Ž , we have
𝐾 1 ξ€· 𝛿 𝑁 𝑓 ξ€Έ ( 𝑒 ) ( + ∞ ) = 0 , ( 2 . 3 6 ) thus 𝑒 ( + ∞ ) = 𝑃 𝑒 + 𝑄 𝛿 𝑁 𝑓 ( 𝑒 ) = 𝑒 ( 0 ) . ( 2 . 3 7 )
From (2.30), we have
𝑀 ξ€· ( 𝑑 ) πœ‘ 𝑑 , 𝑒 ξ…ž ξ€Έ = π‘Ž ξ€· ξ€· 𝐼 βˆ’ 𝑄 βˆ— ξ€Έ 𝛿 𝑁 𝑓 ξ€Έ + 𝐹 ξ€· ξ€· 𝐼 βˆ’ 𝑄 βˆ— ξ€Έ 𝛿 𝑁 𝑓 ξ€Έ ξ€· ( 𝑒 ) ( 𝑑 ) , βˆ€ 𝑑 ∈ ( 0 , + ∞ ) , 𝑀 ( 𝑑 ) πœ‘ ( 𝑑 , 𝑒 ξ…ž ) ξ€Έ ξ…ž = ξ€· 𝐼 βˆ’ 𝑄 βˆ— ξ€Έ 𝛿 𝑁 𝑓 ( 𝑒 ) ( 𝑑 ) , βˆ€ 𝑑 ∈ ( 0 , + ∞ ) . ( 2 . 3 8 )
Obviously 𝐹 ( ( 𝐼 βˆ’ 𝑄 βˆ— ) 𝑁 𝑓 ( 𝑒 ) ) ( + ∞ ) = 0 , from (2.38), we have
𝑀 ξ€· ( 0 ) πœ‘ 0 , 𝑒 ξ…ž ξ€Έ ξ€· ( 0 ) = 𝑀 ( + ∞ ) πœ‘ + ∞ , 𝑒 ξ…ž ξ€Έ . ( + ∞ ) ( 2 . 3 9 )
Since 𝑄 𝛿 𝑁 𝑓 ( 𝑒 ) = 0 , we have 𝑄 βˆ— 𝛿 𝑁 𝑓 ( 𝑒 ) = 0 and
ξ€· 𝑀 ( 𝑑 ) πœ‘ ( 𝑑 , 𝑒 ξ…ž ) ξ€Έ ξ…ž = 𝛿 𝑁 𝑓 ( 𝑒 ) ( 𝑑 ) . ( 2 . 4 0 )
Hence 𝑒 is a solutions of (1.1)-(1.2). This completes the proof.

Lemma 2.5. If 𝑒 is a solution of (1.1)-(1.2), then for any 𝑗 = 1 , … , 𝑁 , there exists an 𝑠 𝑗 ∈ ( 0 , + ∞ ) such that ( 𝑒 𝑗 ) ξ…ž ( 𝑠 𝑗 ) = 0 .

Proof. If it is false, then 𝑒 𝑗 is strictly monotone in ( 0 , + ∞ ) .(i)If 𝑒 𝑗 is strictly decreasing in ( 0 , + ∞ ) , then ( 𝑒 𝑗 ) ( 0 ) > ( 𝑒 𝑗 ) ( + ∞ ) ; it is a contradiction to 𝑒 ( 0 ) = 𝑒 ( + ∞ ) . (ii)If 𝑒 𝑗 is strictly increasing in ( 0 , + ∞ ) , then ( 𝑒 𝑗 ) ( 0 ) < ( 𝑒 𝑗 ) ( + ∞ ) ; it is a contradiction to 𝑒 ( 0 ) = 𝑒 ( + ∞ ) .
This completes the proof.

3. 𝑓 Satisfies Sub-( 𝑝 βˆ’ βˆ’ 1 ) Growth Condition

In this section, we will apply Leray-Schauder's degree to deal with the existence of solutions for (1.1)-(1.2), when 𝑓 satisfies sub-( 𝑝 βˆ’ βˆ’ 1 ) growth condition. Moreover, the asymptotic behavior has been discussed.

Theorem 3.1. Assume that Ξ© is an open bounded set in 𝐢 1 such that the following conditions hold. (10)For each πœ† ∈ ( 0 , 1 ) the problem ξ‚€ | | 𝑒 𝑀 ( π‘Ÿ ) ξ…ž | | 𝑝 ( π‘Ÿ ) βˆ’ 2 𝑒 ξ…ž  ξ…ž ξ€· = πœ† 𝛿 𝑓 π‘Ÿ , 𝑒 , ( 𝑀 ( π‘Ÿ ) ) 1 / ( 𝑝 ( π‘Ÿ ) βˆ’ 1 ) 𝑒 ξ…ž ξ€Έ ( 3 . 1 ) with boundary condition (1.2) has no solution on πœ• Ξ© .(20)The equation πœ” ξ€œ ( π‘Ž ) ∢ = 0 + ∞ 𝛿 𝑓 ( 𝑑 , π‘Ž , 0 ) 𝑑 𝑑 = 0 ( 3 . 2 ) has no solution on πœ• Ξ© ∩ ℝ 𝑁 .(30)The Brouwer degree 𝑑 𝐡