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
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
where
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)
where
(A2)
From (A2), we can see that there exist
Since equation (1.1) is degenerate at points where
Definition 1 A function u is said to be a weak solution of problem (1.1)(1.3) if
for any
Due to the degeneracy of equation (1.1), we consider the following regularized problem:
where
Then we can define a map
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 function
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
Lemma 2Let
where
Proof Multiplying Eq. (2.5) by
and hence
where
Assume that
then
By using the GagliardoNirenberg inequality, we have
with
By inequalities (3.3), (3.4) and the fact that
Let
we have
For Young’s inequality
where
then we have
Here we have used the fact that
Denoting
and combining (3.6), (3.5), we obtain
that is,
From the periodicity of
where
Therefore we conclude that
Since
where
or
where
or
where
Letting
On the other hand, it follows from (3.2) with
By Hölder’s inequality and Sobolev’s theorem, we have
Combined with (3.10), it yields
By Young’s inequality, it follows that
where
which together with (3.9) implies (3.1). The proof is completed. □
Corollary 1There exists a positive constantRindependent ofεsuch that
where
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 existence and uniqueness of the solution of
Lemma 3There exist constants
whereris a positive constant independent ofε.
Proof By contradiction, let
By the periodicity of
and thus
Combining (3.15) with (3.14), we obtain
Let
And also we know that
Multiplying (2.5) by
where
In addition, the assumptions (A1), (A2) give
The above two inequalities imply that
Obviously, we can choose suitably small
Corollary 2There exists a small positive constantrwhich is independent ofεand satisfies
where
Proof Similar to Lemma 3, we can see that there exists a positive constant
Hence the degree is well defined on
Lemma 3 shows that
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
Combining with the regularity results [16] a similar argument to that in [17], we can prove that the limit function of
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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