Open Access

Existence of Solutions and Nonnegative Solutions for Prescribed Variable Exponent Mean Curvature Impulsive System Initialized Boundary Value Problems

  • Guizhen Zhi1,
  • Liang Zhao2,
  • Guangxia Chen3,
  • Xiaopin Liu1 and
  • Qihu Zhang1, 4Email author
Journal of Inequalities and Applications20102010:915739

https://doi.org/10.1155/2010/915739

Received: 10 May 2009

Accepted: 2 January 2010

Published: 23 January 2010

Abstract

This paper investigates the existence of solutions and nonnegative solutions for prescribed variable exponent mean curvature impulsive system initialized boundary value problems. The proof of our main result is based upon Leray-Schauder's degree. The sufficient conditions for the existence of solutions and nonnegative solutions have been given.

1. Introduction

The theory of impulsive differential equations describes processes which experience a sudden change of their state at certain moments. On the Laplacian impulsive differential equations boundary value problems, there are many results (see [15]). Because of the nonlinearity of -Laplacian, the results about -Laplacian impulsive differential equations boundary value problems are rare (see [6]). In [7, 8], the authors discussed the existence of solutions of -Laplacian system impulsive boundary value problems. Recently, the existence and asymptotic behavior of solutions of curvature equations have been studied extensively (see [915]). In [16], the authors generalized the usual mean curvature systems to variable exponent mean curvature systems. In this paper, we consider the existence of solutions and nonnegative solutions for the prescribed variable exponent mean curvature system

(11)

where , with the following impulsive initialized boundary value conditions

(12)
(13)
(14)

where

(15)
are absolutely continuous, where satisfy , is called the variable exponent mean curvature operator, and .

For any , will denote the th component of ; the inner product in will be denoted by will denote the absolute value and the Euclidean norm on . Denote that , , and , , , where , . Denote that is the interior of , . Let

(16)

For any , denote that . Obviously, is a Banach space with the norm ,  and is a Banach space with the norm . In the following, and will be simply denoted by and , respectively. Denote that , and the norm in is .

The study of differential equations and variational problems with variable exponent conditions is a new and interesting topic. For the applied background on this kind of problems we refer to [1719]. Many results have been obtained on this kind of problems, for example, [2035]. If (a constant) and (a constant), then (1.1) is the well-known mean curvature system. Since problems with variable exponent growth conditions are more complex than those with constant exponent growth conditions, many methods and results for the latter are invalid for the former; for example, if is a bounded domain, the Rayleigh quotient

(17)

is zero in general, and only under some special conditions (see [25]), but the fact that is very important in the study of -Laplacian problems.

In this paper, we investigate the existence of solutions for the prescribed variable exponent mean curvature impulsive differential system initialized boundary value problems; the proof of our main result is based upon Leray-Schauder's degree. This paper was motivated by [6, 13, 36].

Let , then the function is assumed to be Caratheodory; by this we mean that

(i)for almost every the function is continuous,

(ii)for each the function is measurable on ,

(iii)for each there is a such that, for almost every and every with , , one has
(18)

We say a function is a solution of (1.1) if with absolutely continuous on ,   , which satisfies (1.1) on .

This paper is divided into three sections; in the second section, we present some preliminary. Finally, in the third section, we give the existence of solutions and nonnegative solutions for system (1.1)–(1.4).

2. Preliminary

In this section, we will do some preparation.

Lemma 2.1 (see [16]).

is a continuous function and satisfies the following.
(i)For any is strictly monotone, that is,
(21)
(ii)For any fixed , is a homeomorphism from to
(22)
For any , denote by the inverse operator of , then
(23)
Let one now consider the following simple problem:
(24)
with the following impulsive boundary value conditions:
(25)

where ; .

Obviously, implies that If is a solution of (2.4) with (2.5), by integrating (2.4) from to , then one finds that
(26)
Define , and operator as
(27)
From the definition of , one can see that
(28)
Denote that
(29)
By (2.6), one has
(210)
Denote that with the norm
(211)

then is a Banach space.

Let one define the nonlinear operator as
(212)

Lemma 2.2.

The operator is continuous and sends closed equiintegrable subsets of into relatively compact sets in .

Proof.

It is easy to check that , for all . Since
(213)

it is easy to check that is a continuous operator from to .

Let now be a closed equiintegrable set in , then there exists , such that, for any ,
(214)

We want to show that is a compact set.

Let is a sequence in , then there exists a sequence such that . For any , we have that
(215)
Hence the sequence is uniformly bounded and equicontinuous. By Ascoli-Arzela theorem, there exists a subsequence of (which we rename the same) that is convergent in . Then the sequence
(216)
is convergent according to the norm in . Since
(217)

according to the continuity of , we can see that is convergent in . Thus we conclude that is convergent in . This completes the proof.

We denote that is the Nemytski operator associated to defined by

(218)

Define as

(219)

where .

Lemma 2.3.

is a solution of (1.1)–(1.4) if and only if is a solution of the following abstract equation:
(220)

where , .

Proof.
  1. (i)
    If is a solution of (1.1)–(1.4), since implies that , by integrating (1.1) from to , then we find that
    (221)
     
Thus
(222)
Hence is a solution of (2.20).
  1. (ii)
    If is a solution of (2.20), then it is easy to see that (1.2) is satisfied. Let , then we have
    (223)
     
From (2.20) we also have
(224)
(225)
From (2.24), we can see that (1.3) is satisfied. Let , from (2.24), then we have
(226)
Since we must have ; thus,
(227)

Hence is a solution of (1.1)–(1.4). This completes the proof.

3. Main Results and Proofs

In this section, we will apply Leray-Schauder's degree to deal with the existence of solutions for (1.1)–(1.4).

We assume that

(H) for all

Theorem 3.1.

If (H ) is satisfied, then is an open bounded set in such that the following conditions hold.

(1) For any the mapping belongs to .

(2) For each , the problem

(31)

has no solution on .

Then (1.1)–(1.4) has at least one solution on .

Proof.

Denote that
(32)
For any , define as
(33)

Denote that

We know that (1.1)–(1.4) has the same solutions of
(34)

Since is Caratheodory, it is easy to see that is continuous and sends bounded sets into equiintegrable sets. According to Lemma 2.2, we can conclude that is compact continuous on for any . We assume that for (3.4) does not have a solution on ; otherwise, we complete the proof. Now from hypothesis (2 ), it follows that (3.4) has no solution for , .

When , (3.1) is equivalent to the following usual problem:
(35)

where obviously is the unique solution to this problem.

Since , we have proved that (3.4) has no solution , on , then we get that, for each , Leray-Schauder's degree is well defined. From the homotopy invariant property of that degree, we have
(36)

This completes the proof.

In the following, we will give an application of Theorem 3.1.

Denote that and

(37)

Obviously, is an open subset of .

Assume that

(H) , where is a positive parameter, and is Caratheodory.

(H) ,   for all .

Theorem 3.2.

If are satisfied, then problem (1.1)–(1.4) has at least one solution on , when positive parameter is small enough.

Proof.

Let one consider the problem
(38)

where is defined in (3.3).

Obviously, is a solution of (1.1)–(1.4) if and only if is a solution of the abstract equation (3.8) when . We only need to prove that the conditions of Theorem 3.1 are satisfied.

(1 ) When positive parameter is small enough, for any , we can see that the mapping belongs to .

(2 ) We shall prove that for each the problem
(39)

has no solution on .

If it is false, then there exists a and is a solution of (3.8). Then there exists an such that .
  1. (i)
    Suppose that , then . For any since according to (H ) and (1.2), we have
    (310)
     
It is a contradiction.
  1. (ii)

    Suppose that , . This implies that for some .

     
Denote that
(311)
Since we have
(312)
According to (H )-(H ), when positive parameter is small enough, we have
(313)

It is a contradiction.

Summarizing this argument, for each the problem (3.8) with (1.4) has no solution on .

Since and is the unique solution of , then the Leray-Schauder's degree
(314)

This completes the proof.

In the following, we will discuss the existence of nonnegative solutions of (1.1)–(1.4). For any , the notation means that for every .

Assume the following

(H) where is a positive parameter, and
(315)

where , and for all .

(H) and for all , .

(H) .

(H) satisfies for all .

Theorem 3.3.

If H H are satisfied, then the problem (1.1)–(1.4) has a nonnegative solution, when positive parameter is small enough.

Proof.

From Theorem 3.2, we can get the existence of solutions of (1.1)–(1.4). If is a solution of (1.1)–(1.4), according to (1.4) and (H ), then we have
(316)
Obviously
(317)
When , we have
(318)
then we can see that there exists a such that when . Thus for any . Thus is increasing in , that is, for any with . Since , it is easy to see that for any . From (3.17) and (H ), we can easily see that
(319)
From (H ), we can see that
(320)
Similarly, we can see that
(321)
Repeating the step, we can see that
(322)

Hence is nonnegative. This completes the proof.

Declarations

Acknowledgments

This paper is partly supported by the National Science Foundation of China (10701066, 10926075 and 10971087), China Postdoctoral Science Foundation funded project (20090460969), the Natural Science Foundation of Henan Education Committee (2008-755-65), and the Natural Science Foundation of Jiangsu Education Committee (08KJD110007).

Authors’ Affiliations

(1)
Department of Mathematics and Information Science, Zhengzhou University of Light Industry
(2)
Academic Administration, Henan Institute of Science and Technology
(3)
School of Mathematics and Informatics, Henan Polytechnic University
(4)
School of Mathematics and Statistics, Huazhong Normal University

References

  1. Jiao J, Chen L, Li L: Asymptotic behavior of solutions of second-order nonlinear impulsive differential equations. Journal of Mathematical Analysis and Applications 2008, 337(1):458–463. 10.1016/j.jmaa.2007.04.021MathSciNetView ArticleMATHGoogle Scholar
  2. Liu L, Hu L, Wu Y: Positive solutions of two-point boundary value problems for systems of nonlinear second-order singular and impulsive differential equations. Nonlinear Analysis: Theory, Methods & Applications 2008, 69(11):3774–3789. 10.1016/j.na.2007.10.012MathSciNetView ArticleMATHGoogle Scholar
  3. Liu X, Guo D: Periodic boundary value problems for a class of second-order impulsive integro-differential equations in Banach spaces. Journal of Mathematical Analysis and Applications 1997, 216(1):284–302. 10.1006/jmaa.1997.5688MathSciNetView ArticleMATHGoogle Scholar
  4. Shen J, Wang W: Impulsive boundary value problems with nonlinear boundary conditions. Nonlinear Analysis: Theory, Methods & Applications 2008, 69(11):4055–4062. 10.1016/j.na.2007.10.036MathSciNetView ArticleMATHGoogle Scholar
  5. Yao M, Zhao A, Yan J: Periodic boundary value problems of second-order impulsive differential equations. Nonlinear Analysis: Theory, Methods & Applications 2009, 70(1):262–273. 10.1016/j.na.2007.11.050MathSciNetView ArticleMATHGoogle Scholar
  6. Cabada A, Tomeček J: Extremal solutions for nonlinear functional -Laplacian impulsive equations. Nonlinear Analysis: Theory, Methods & Applications 2007, 67(3):827–841. 10.1016/j.na.2006.06.043MathSciNetView ArticleMATHGoogle Scholar
  7. Zhang Q, Liu X, Qiu Z: Existence of solutions for weighted -Laplacian impulsive system periodic-like boundary value problems. Nonlinear Analysis: Theory, Methods & Applications 2009, 71(9):3596–3611. 10.1016/j.na.2009.02.043MathSciNetView ArticleMATHGoogle Scholar
  8. Zhang Q, Qiu Z, Liu X: Existence of solutions and nonnegative solutions for weighted -Laplacian impulsive system multi-point boundary value problems. Nonlinear Analysis: Theory, Methods & Applications 2009, 71(9):3814–3825. 10.1016/j.na.2009.02.040MathSciNetView ArticleMATHGoogle Scholar
  9. Filippucci R: Entire radial solutions of elliptic systems and inequalities of the mean curvature type. Journal of Mathematical Analysis and Applications 2007, 334(1):604–620. 10.1016/j.jmaa.2007.01.005MathSciNetView ArticleMATHGoogle Scholar
  10. Greco A: On the existence of large solutions for equations of prescribed mean curvature. Nonlinear Analysis: Theory, Methods & Applications 1998, 34(4):571–583. 10.1016/S0362-546X(97)00556-7MathSciNetView ArticleMATHGoogle Scholar
  11. Ivochkina NM: The Dirichlet problem for the curvature equation of order . Leningrad Mathematical Journal 1991, 2: 631–654.MathSciNetMATHGoogle Scholar
  12. Lancaster KE, Siegel D: Existence and behavior of the radial limits of a bounded capillary surface at a corner. Pacific Journal of Mathematics 1996, 176(1):165–194.MathSciNetView ArticleMATHGoogle Scholar
  13. Pan H: One-dimensional prescribed mean curvature equation with exponential nonlinearity. Nonlinear Analysis: Theory, Methods & Applications 2009, 70(2):999–1010. 10.1016/j.na.2008.01.027MathSciNetView ArticleMATHGoogle Scholar
  14. Takimoto K: Solution to the boundary blowup problem for -curvature equation. Calculus of Variations and Partial Differential Equations 2006, 26(3):357–377. 10.1007/s00526-006-0011-7MathSciNetView ArticleMATHGoogle Scholar
  15. Trudinger NS: The Dirichlet problem for the prescribed curvature equations. Archive for Rational Mechanics and Analysis 1990, 111(2):153–179. 10.1007/BF00375406MathSciNetView ArticleMATHGoogle Scholar
  16. Zhang Q, Qiu Z, Liu X: Existence of solutions for prescribed variable exponent mean curvature system boundary value problems. Nonlinear Analysis: Theory, Methods & Applications 2009, 71(7–8):2964–2975. 10.1016/j.na.2009.01.190MathSciNetView ArticleMATHGoogle Scholar
  17. Chen Y, Levine S, Rao M: Variable exponent, linear growth functionals in image restoration. SIAM Journal on Applied Mathematics 2006, 66(4):1383–1406. 10.1137/050624522MathSciNetView ArticleMATHGoogle Scholar
  18. Růžička M: Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics. Volume 1748. Springer, Berlin, Germany; 2000:xvi+176.MATHGoogle Scholar
  19. Zhikov VV: Averaging of functionals of the calculus of variations and elasticity theory. Mathematics of the USSR. Izvestiya 1987, 29: 33–36. 10.1070/IM1987v029n01ABEH000958View ArticleMATHGoogle Scholar
  20. Fan X-L, Wu H-Q, Wang F-Z: Hartman-type results for -Laplacian systems. Nonlinear Analysis: Theory, Methods & Applications 2003, 52(2):585–594. 10.1016/S0362-546X(02)00124-4MathSciNetView ArticleMATHGoogle Scholar
  21. Fan X-L, Zhao D: Regularity of minimum points of variational integrals with continuous -growth conditions. Chinese Annals of Mathematics. Series A 1996, 17(5):557–564.MathSciNetMATHGoogle Scholar
  22. Fan X-L, Zhao D: On the spaces and . Journal of Mathematical Analysis and Applications 2001, 263(2):424–446. 10.1006/jmaa.2000.7617MathSciNetView ArticleMATHGoogle Scholar
  23. Fan X-L, Zhang Q-H: Existence of solutions for -Laplacian Dirichlet problem. Nonlinear Analysis: Theory, Methods & Applications 2003, 52(8):1843–1852. 10.1016/S0362-546X(02)00150-5MathSciNetView ArticleMATHGoogle Scholar
  24. Fan X-L, Zhao YZ, Zhang QH: A strong maximum principle for -Laplace equations. Chinese Journal of Contemporary Mathematics 2003, 24(4):495–500.MathSciNetMATHGoogle Scholar
  25. Fan X-L, Zhang QH, Zhao D: Eigenvalues of -Laplacian Dirichlet problem. Journal of Mathematical Analysis and Applications 2005, 302(2):306–317. 10.1016/j.jmaa.2003.11.020MathSciNetView ArticleMATHGoogle Scholar
  26. El Hamidi A: Existence results to elliptic systems with nonstandard growth conditions. Journal of Mathematical Analysis and Applications 2004, 300(1):30–42. 10.1016/j.jmaa.2004.05.041MathSciNetView ArticleMATHGoogle Scholar
  27. Kováčik O, Rákosník J: On spaces and . Czechoslovak Mathematical Journal 1991, 41(4):592–618.MathSciNetMATHGoogle Scholar
  28. Marcellini P: Regularity and existence of solutions of elliptic equations with -growth conditions. Journal of Differential Equations 1991, 90(1):1–30. 10.1016/0022-0396(91)90158-6MathSciNetView ArticleMATHGoogle Scholar
  29. Mihăilescu M, Rădulescu V: Continuous spectrum for a class of nonhomogeneous differential operators. Manuscripta Mathematica 2008, 125(2):157–167. 10.1007/s00229-007-0137-8MathSciNetView ArticleMATHGoogle Scholar
  30. Musielak J: Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics. Volume 1034. Springer, Berlin, Germany; 1983:iii+222.Google Scholar
  31. Samko SG: Density in the generalized Sobolev spaces . Doklady Akademii Nauk. Rossiĭskaya Akademiya Nauk 1999, 369(4):451–454.MathSciNetMATHGoogle Scholar
  32. Zhang QH: A strong maximum principle for differential equations with nonstandard -growth conditions. Journal of Mathematical Analysis and Applications 2005, 312(1):24–32. 10.1016/j.jmaa.2005.03.013MathSciNetView ArticleMATHGoogle Scholar
  33. Zhang QH: Existence of solutions for -Laplacian equations with singular coefficients in . Journal of Mathematical Analysis and Applications 2008, 348(1):38–50. 10.1016/j.jmaa.2008.06.026MathSciNetView ArticleMATHGoogle Scholar
  34. Zhang Q, Liu X, Qiu Z: The method of subsuper solutions for weighted -Laplacian equation boundary value problems. Journal of Inequalities and Applications 2008, 2008:-18.MathSciNetMATHGoogle Scholar
  35. Zhang Q, Qiu Z, Liu X: Existence of solutions for a class of weighted -Laplacian system multipoint boundary value problems. Journal of Inequalities and Applications 2008, 2008:-18.MathSciNetMATHGoogle Scholar
  36. Manásevich R, Mawhin J: Periodic solutions for nonlinear systems with -Laplacian-like operators. Journal of Differential Equations 1998, 145(2):367–393. 10.1006/jdeq.1998.3425MathSciNetView ArticleMATHGoogle Scholar

Copyright

© Guizhen Zhi et al. 2010

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