Existence of Solutions and Nonnegative Solutions for Prescribed Variable Exponent Mean Curvature Impulsive System Initialized Boundary Value Problems
© Guizhen Zhi et al. 2010
Received: 10 May 2009
Accepted: 2 January 2010
Published: 23 January 2010
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.
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 [1–5]). Because of the nonlinearity of -Laplacian, the results about -Laplacian impulsive differential equations boundary value problems are rare (see ). 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 [9–15]). In , 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
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
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 [17–19]. Many results have been obtained on this kind of problems, for example, [20–35]. 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
is zero in general, and only under some special conditions (see ), 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].
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).
In this section, we will do some preparation.
Lemma 2.1 (see ).
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
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 , .
This completes the proof.
In the following, we will give an application of Theorem 3.1.
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.
It is a contradiction.
This completes the proof.
Assume the following
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).
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