Abstract
In this paper, we study a periodic pLaplacian equation with nonlocal terms and Neumann boundary conditions. We establish the existence of time periodic solutions of the pLaplacianNeumann problem by the theory of LeraySchauder degree.
1 Introduction
In this paper, we consider the periodic boundary problem for a pLaplacian equation of the following form:
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.
The model as (1.1) was first studied by Allegretto and Nistri. In [1] they studied the existence of nontrivial nonnegative periodic solutions and optimal control for the following equation:
with Dirichlet boundary value conditions. Later, many mathematical researchers studied extended forms of this kind of equation. For example, in [27], the authors considered some nonlinear diffusion equations with nonlocal terms such as the porous equation with , the pLaplacian equation with and the doubly degenerate parabolic equation. All these problems are the Dirichlet boundary value conditions, and these boundary conditions describe that the boundary we consider in this model is lethal to the species. Moreover, the methods in these papers are all based on the theory of LeraySchauder degree. However, there are few results on degenerate periodic parabolic equations with nonlocal terms and Neumann boundary conditions. Recently, in [8], Wang and Yin considered the following periodic Neumann boundary value problem:
where . By the parabolic regularized method and the theory of LeraySchauder 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 [913] 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 pLaplacian Neumann problem (1.1)(1.3). In [14], the authors studied equation (1.1) with the Dirichlet boundary value condition. Compared with the Dirichlet boundary value condition in [14], 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 LeraySchauder 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.
2 Preliminaries
In this paper, we assume that
(A1) is a bounded continuous functional satisfying
where are constants independent of u, , ;
(A2) and satisfies that , where denotes the set of functions which are continuous in and of Tperiodic with respect to t.
From (A2), we can see that there exist , , such that
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.
Definition 1 A function u is said to be a weak solution of problem (1.1)(1.3) if and satisfies
Due to the degeneracy of equation (1.1), we consider the following regularized problem:
where and ε is a sufficiently small positive constant. The desired solution will be obtained as the limit point of the solutions of problem (2.2)(2.4). In the following, we introduce a map by the following problem:
Then we can define a map with . Applying classical estimates (see [15]), we can know that is bounded by and is Hölder continuous in . Then by the ArzelaAscoli 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 [14], we can obtain the nonnegativity of the solution of problem (2.2)(2.4).
Lemma 1If a nontrivial functionsolves, then
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.
Lemma 2Let, be a nonnegative periodic solution solving, then there exists a constantindependent ofλ, εsuch that
Proof Multiplying Eq. (2.5) by () and integrating over Ω, we have
and hence
where () are positive constants independent of and m.
then , . For convenience, we denote by C a positive constant independent of k and m, which may take different values. From (3.2) we obtain
By using the GagliardoNirenberg inequality, we have
with
By inequalities (3.3), (3.4) and the fact that , we obtain the following differential inequality:
Let
we have
For Young’s inequality
then we have
Here we have used the fact that for some r independent of k. In fact, it is easy to verify that
Denoting
and combining (3.6), (3.5), we obtain
that is,
From the periodicity of , we know that there exists at which reaches its maximum and thus the lefthand side of (3.8) vanishes. Then we obtain
where
Therefore we conclude that
Since and and are bounded, we have
where is a positive constant independent of k. As implies that and , we get
or
or
where
On the other hand, it follows from (3.2) with that
By Hölder’s inequality and Sobolev’s theorem, we have
Combined with (3.10), it yields
By Young’s inequality, it follows that
where () are constants independent of u. Taking the periodicity of u into account, we infer from (3.12) that
which together with (3.9) implies (3.1). The proof is completed. □
Corollary 1There exists a positive constantRindependent ofεsuch that
whereis a ball centered at the origin with radiusRin.
Proof It follows from Lemma 2 that there exists a positive constant R independent of ε such that
Hence the degree is well defined on . From the homotopy invariance of the LeraySchauder degree, we can see that
From the existence and uniqueness of the solution of , we have . That is, . The proof is completed. □
Lemma 3There exist constantsandsuch that for any, , admits no nontrivial solutionsatisfying
whereris a positive constant independent ofε.
Proof By contradiction, let be a nontrivial solution of satisfying . For any given , multiplying (2.5) by and integrating over , we obtain
By the periodicity of , the first term on the lefthand side in (3.14) is zero. As in the proof of Lemma 2.2 of [14], the second term on the lefthand side in (3.14) can be rewritten as
and thus
Combining (3.15) with (3.14), we obtain
Let be the first eigenvalue of the pLaplacian equation on Ω with zero boundary conditions and be the corresponding eigenfunction. We have
And also we know that can be strictly positive in the subfield . Taking , we have
Multiplying (2.5) by and integrating over , from the assumption , we have
where and denotes the Lebesgue measure of the domain Ω. Combining (3.18) with (3.17), we obtain
In addition, the assumptions (A1), (A2) give
The above two inequalities imply that
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. □
Corollary 2There exists a small positive constantrwhich is independent ofεand satisfiessuch that
whereis a ball centered at the origin with radiusrin.
Proof Similar to Lemma 3, we can see that there exists a positive constant independent of ε such that
Hence the degree is well defined on . From the homotopy invariance of the LeraySchauder degree, we can see that
Lemma 3 shows that admits no nontrivial solution in and it is also easy to see that is not a solution of . So, we have , that is,
The proof is completed. □
Theorem 1If assumptions (A1) and (A2) hold, then problem (1.1)(1.3) admits a nontrivial nonnegative periodic solutionu.
Proof Using Corollary 1 and Corollary 2, we know that
where , is a ball centered at the origin with radius ξ in , R and r are positive constants and . By the theory of the LeraySchauder degree and Lemma 1, we can conclude that problem (2.2)(2.4) admits a nontrivial nonnegative periodic solution . By Lemma 3 and a similar method to that in [14], we can obtain
Combining with the regularity results [16] a similar argument to that in [17], we can prove that the limit function of is a nonnegative nontrivial periodic solution of problem (1.1)(1.3). □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
RH, JS and BW carried out the proof of the main part of this article, BW corrected the manuscript and participated in its design and coordination. All authors have read and approved the final manuscript.
Acknowledgements
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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