Existence of periodic solutions of a p-Laplacian-Neumann problem
© Hameed et al.; licensee Springer 2013
Received: 6 March 2013
Accepted: 7 June 2013
Published: 23 July 2013
In this paper, we study a periodic p-Laplacian equation with nonlocal terms and Neumann boundary conditions. We establish the existence of time periodic solutions of the p-Laplacian-Neumann problem by the theory of Leray-Schauder degree.
where , Ω is a bounded domain in with smooth boundary ∂ Ω, denotes the outward normal derivative on ∂ Ω, . This problem is motivated by models which have been proposed for some problems in mathematical biology. The function u represents the spatial densities of the species at ; the diffusion term represents the effect of dispersion in the habitat, which models a tendency to avoid crowding, and the speed of the diffusion is slow since ; the term models the contribution of the population supply due to births and deaths; the Neumann boundary conditions model the trend of the biology population to survive on the boundary. Assumptions of m, will be introduced in the next section.
where . By the parabolic regularized method and the theory of Leray-Schauder degree, they established the existence of nontrivial nonnegative periodic solutions. Also, there are many works about reaction diffusion equations without nonlocal term; one can see [9–13] and the references therein, and the boundary value condition and research method are different from our work.
In this paper, we consider the periodic solution of p-Laplacian Neumann problem (1.1)-(1.3). In , the authors studied equation (1.1) with the Dirichlet boundary value condition. Compared with the Dirichlet boundary value condition in , the Neumann boundary value condition causes an additional difficulty in establishing some a priori estimates. On the other hand, different from that in the case of the Dirichlet boundary value condition, the standard regularized problem of problem (1.1)-(1.3) is not well posed, and thus a modified regularized problem for (1.1)-(1.3) is considered. In addition, we will make use of the Moser iterative method to establish the a priori upper bound of the solution of the regularized problem. By the theory of Leray-Schauder degree, we prove that this modified problem admits nontrivial nonnegative periodic solutions. Then, by passing to a limit process, we obtain the existence of nontrivial nonnegative periodic solutions of problem (1.1)-(1.3). In the process of proving the main results, the nonlocal term, which reflects the reality of the model (1.1), will cause a difficulty when we establish a lower bound estimate of the maximum modulus of the solution of the regularized problem. Otherwise, we can use the method of upper and lower solution to prove the existence of periodic solutions. At last, the existence theorem shows that the spatial densities of the species are periodic under the case of nonlinear diffusion.
This paper is organized as follows. In Section 2, we show some necessary preliminaries including the modified regularized problem. In Section 3, we establish some necessary a priori estimations of the solution of the modified regularized problem. Then we obtain the main results of this paper.
In this paper, we assume that
- (A1)is a bounded continuous functional satisfying
where are constants independent of u, , ;
and satisfies that , where denotes the set of functions which are continuous in and of T-periodic with respect to t.
Since equation (1.1) is degenerate at points where , problem (1.1)-(1.3) has no classical solutions in general, so we focus on the discussion of weak solutions in the sense of the following.
for any with .
Then we can define a map with . Applying classical estimates (see ), we can know that is bounded by and is Hölder continuous in . Then by the Arzela-Ascoli theorem, the map G is compact. So, the map is a compact continuous map. Let , where , we can see that the nonnegative solution of problem (2.2)-(2.4) is also a nonnegative solution solving . So, we will study the existence of nonnegative fixed points of the map instead of the nonnegative solutions of problem (2.2)-(2.4).
3 Proof of the main results
First, by the same method as in , we can obtain the nonnegativity of the solution of problem (2.2)-(2.4).
In the following, by the Moser iterative technique, we will show the a priori estimate for the upper bound of nonnegative periodic solutions of problem (2.5)-(2.7). Here and below we denote by () the norm.
where () are positive constants independent of and m.
which together with (3.9) implies (3.1). The proof is completed. □
where is a ball centered at the origin with radius R in .
From the existence and uniqueness of the solution of , we have . That is, . The proof is completed. □
where r is a positive constant independent of ε.
Obviously, we can choose suitably small and such that for any , , the inequality (3.21) does not hold. It is a contradiction. The proof is completed. □
where is a ball centered at the origin with radius r in .
The proof is completed. □
Theorem 1 If assumptions (A1) and (A2) hold, then problem (1.1)-(1.3) admits a nontrivial nonnegative periodic solution u.
This work is partially supported by the National Science Foundation of China (11271100, 11126222), the Fundamental Research Funds for the Central Universities (Grant No. HIT. NSRIF. 2011006), the Natural Sciences Foundation of Heilongjiang Province (QC2011C020) and also the 985 project of Harbin Institute of Technology.
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