The existence of positive solutions to an elliptic system with nonlinear boundary conditions
Boundary Value Problemsvolume 2013, Article number: 159 (2013)
In this paper, we consider the following system:
where Ω is a bounded domain in () with smooth boundary, is the outer normal derivative and are positive and continuous functions. Under certain assumptions on and , but without the usual (AR) condition, we prove that the problem has at least one positive strong pair solution (see Definition 1.4 below) by applying a linking theorem for strong indefinite functional.
MSC:35A01, 35J20, 35J25.
1 Introduction and main result
In this paper, we mainly study the following system:
where Ω is a bounded domain in () with smooth boundary, is the outer normal derivative and are positive and continuous functions.
Existence results for nonlinear elliptic systems have received a lot of interest in recent years (see [1–12]), particularly when the nonlinear term appears as a source in the equation, complemented with Dirichlet boundary conditions. To our knowledge, about the system with nonlinear boundary conditions, there are not many results. Here we refer to [9, 13, 14].
In , Li and one of the authors considered
Under some given conditions, we proved that (1.2) had at least one positive solution pair .
In , Bonder, Pinasco, Rossi studied
They assumed that H satisfied the following conditions:
() The Ambrosetti-Rabinowitz type condition: For R large, if ,
where and with
When , they also assumed
() , .
They obtained infinitely many nontrivial solutions of (1.3) under the assumptions () to () by using variational arguments and a fountain theorem. Note that (1.4) implies
(see Lemma 1.1 in ). Therefore it is not difficult to verify that any (PS) sequence (or (C) c sequence) of the corresponding functional is bounded in some suitable space.
The crucial part in the nonlinear boundary conditions case is to find the proper functional setting for (1.1) that allows us to treat our problem variationally. We accomplish this by defining a self-adjoint operator that takes into account the boundary conditions together with the equations and considering its fractional powers that satisfy a suitable ‘integration by parts’ formula. In order to obtain nontrivial solutions, we use a linking theorem (see ).
The assumptions we impose on and are as follows:
(H1) with for any ,
(H2) uniformly in .
(H3) There is a positive constant such that
where and satisfy
(H4) uniformly in , where , .
(H5) For all , or , , there are two positive constants , such that
where , with for any , .
Remark 1.1 By (1.11), there exist l and m with , such that
Remark 1.2 (H5) was first introduced by Miyagaki and Souto in . A typical pair of functions , , , ; , satisfy (H1) to (H5). However, the pair of functions , , ; , satisfy (H5) but do not satisfy the usual (AR) condition and () in this paper.
Remark 1.3 The assumptions we impose on f and g are different from the assumptions in . To our best knowledge, it is the first time the group assumptions have been used to deal with a system with nonlinear boundary conditions.
In order to state our main result, first we give a definition.
Definition 1.4 We say that is a strong solution of (1.1) if
and satisfies (1.1) a.e. in Ω.
Our main results is as follows.
Theorem 1.5 Let (H1)-(H5) hold. Then system (1.1) possesses at least one positive strong solution pair .
The main difficulties to deal with system (1.1) consist in at least three aspects. Firstly, due to the type of growth of the functions f and g, we cannot work with the usual , and then we need fractional Sobolev spaces. Secondly, although we have a variational problem, the functional associated to it always has a strong indefinite quadratic part. So, the functional possesses no mountain-pass structure but the linking geometric structure, which is more complicated to handle. Thirdly, as we do not assume that the functions f and g satisfy the (AR) conditions, it is much more difficult to show that any (C) c sequence is uniformly bounded in E (see Section 2).
To prove Theorem 1.5, we try to find a critical point of the functional Φ (see (2.5)) in E. We prove that Φ has a linking geometric structure and use a linking theorem under (C) c condition (see Theorem 2.1 in ) to get a (C) c -sequence of Φ. The main difficulty now will be to prove that is uniformly bounded in E without the (AR) condition. Then we prove that any (C) c -sequence of Φ is bounded. To overcome this difficulty, we use some techniques used in [12, 16] for which the assumptions (H4), (H5) play important roles. As is bounded, then we can prove that has a subsequence which converges to a nontrivial critical point of Φ. Hence, by the strong maximum principle, we can prove that the pair solution is positive.
The paper is organized as follows. In Section 2, we give some preliminaries. We prove our main result in Section 3.
2 Some preliminaries
In this section we mainly give some preliminaries which will be used in Section 3. We follow the structure in .
Throughout this paper, we consider the space which is a Hilbert space with the inner product, which we denote by , given by
Now we let be the operator defined by
where . It is not difficult to verify that is dense in . Note that A is invertible with its inverse given by
where u is the solution of
By standard regularity (see ), it follows that is bounded and compact. Hence, . Therefore, in order to see that A (hence ) is self-adjoint, it suffices to prove that A is symmetric (, p.512). In fact, for , applying Green’s formula, we obtain
Hence A is symmetric. Also we can check that A (and so ) is positive. For any and by Green’s formula again, we have
Hence there is a sequence of eigenvalues with eigenfunctions satisfying and , ,
Now we consider the following fractional powers of A, i.e., for ,
where . Let , which is a Hilbert space under the inner product
Note that . Indeed, if we define by
Since , we have . Therefore,
The following compact result will be useful later.
Proposition 2.1 (Theorem 2.1, )
Given and so that , the inclusion map is well defined and bounded. Moreover, if , then the inclusion is compact.
Denote , where , l, m are the same as in Remark 1.1 and define by
Associated to B, we have the quadratic form
It is easy to see (one can refer to ) that the bounded self-adjoint operator defined by has exactly two eigenvalues +1 and −1, and that the corresponding eigenvalues and are given by
where we use the notation . Then . The spaces and are orthogonal with respect to the bilinear B, that is,
Moreover, we have
if . We see also that for , with , and
From (1.10), Remark 1.1 and Proposition 2.1, we can define the functional as
Lemma 2.2 The functional ℋ defined by (2.4) is of class and its derivative is given by
Moreover, is compact.
Proof From (1.10), Hölder’s inequality and Proposition 2.1, we have
Similarly, we have
Hence is well defined and bounded in E. A standard argument yields that ℋ is Fréchet differentiable with continuous. By Proposition 2.1 we know that is compact (see  for the details). □
Now we define the functional for (1.1) given by
Moreover, Φ is class .
Definition 2.3 We say that is an -weak solution of (1.1) if z is a critical point of Φ. In other words, for every , we have
Now we give a regularity result of an weak solution.
Proposition 2.4 If is an -weak solution of (1.1), then , and
In other words, is a strong solution of (1.1).
Proof Although the proof is only needed to make some minor modifications as that of Theorem 2.2 in , for the readers’ convenience, we give its detailed proof.
Let us consider in (2.6), then
for all .
If we take , then we have
On the other hand, by (1.10) and Proposition 2.1, we have
i.e., . Then from basic elliptic theory (Theorem 9.9, p.9, ) there exists one function such that
Then we get
From (2.10), (2.11) and (2.12), we have
which implies that . We have gotten that . Finally, since , we conclude that u satisfies (2.7). We can make the same argument for v. □
3 The proof of our main result
In this section, we mainly want to prove Theorem 1.5. First we present a linking theorem from . Then we prove that it can be applied to our functional setting stated in Section 2.
Suppose that , satisfy the assumptions (H1)-(H3), then it is easy to see that for any there is a such that for we have
Since when , is a solution of (1.1). So we are interested in nontrivial and nonnegative solutions of (1.1).
Recall that is called a Palais-Smale sequence of a functional I on E at level c ((PS) c -sequence for short) if and in as . If and in as , then will be called a Cerami sequence at level c ((C) c -sequence for short). A standard way to prove the existence of a positive solution to (1.1) is to get a (PS) c or (C) c sequence for Φ and then to prove that the sequence converges to a solution to (1.1). In this paper, we want to get a (C) c sequence by a linking theorem (Theorem 2.1, in ). So, we need to recall some terminology (see, e.g., [11, 22]).
Let be a closed separable subspace of a Hilbert space H with the norm and let . For , we shall write , where . On H we define a new norm
where is a total orthonormal sequence in . The topology generated by will be called the τ-topology. Recall from  that a homotopy , where , is called admissible if:
h is τ-continuous, i.e., in τ-topology as whenever in τ-topology and as .
g is τ-locally finite-dimensional, i.e., for each , there is a neighborhood U of in the product topology of and such that is contained in a finite-dimensional subspace of H.
Admissible maps are defined similarly. Recall also that admissible maps and homotopies are necessarily continuous, and on bounded subsets of H the τ-topology coincides with the product topology of and .
Let , and and define
Proposition 3.1 (Theorem 2.1, )
Let be a separable Hilbert space with orthogonal to . Suppose that
, where is bounded below, weakly sequentially lower semi-continuous and is weakly sequentially continuous.
There exist , , and such that and .
Then there exists a (C) c -sequence for Φ, where
For fixed and , let
Lemma 3.2 There exist and such that .
Proof For any , , we know that or, equivalently, . By (3.2) and Proposition 2.1, we have
Since , we have if is small enough,
for some . □
Lemma 3.3 For the r given by Lemma 3.2 and any with , there exists such that , where .
Proof If , then with either , or , .
If , then we have , and
since for any .
Assume that . We argue by contradiction. Suppose that there exists a sequence , , , , such that . If , then by the definitions of and , we have
Denote , . Then
Since for any , by (3.3) we know that .
On the other hand, , which implies that for some and as , where ⇀ denotes the weak convergence in E.
If , then from (3.3) we get
which is impossible.
If , since and as , it follows that . If is such that , we have
Similarly, if , we have
Since and , we get
as . Hence, by Proposition 2.1 we may assume, passing to a subsequence, that
as . By (3.8), (3.9) and (H4), taking limit in (3.10), using Fatou’s lemma and the fact that , we obtain
which is impossible, thus the lemma is proved. □
Lemma 3.4 If is a (C) c -sequence of Φ, then is bounded in E.
Proof Suppose that is a (C) c sequence for Φ, that is,
which shows that
where as .
We suppose, by contradiction, that
and let . Then with
By Proposition 2.1, contains a subsequence, denoted again by such that we may assume that
Let . Then we have
and (3.7) implies that
We may assume, without loss of generality, that
By (H4), we see that
This means that
By (H4), there is an such that
for any and with . Since is continuous on , there is an such that
for . From (3.11) and (3.12), we see that there is a constant C, such that for any , we have
which shows that
This means that
Since by (3.6) we have that
which shows that
Since , we have
We claim that .
If , then by Fatou’s lemma, (H4) and Hölder’s inequality, we get
which is impossible.
This shows that
Hence a.e. in ∂ Ω.
Since is continuous in , there exists (), such that
As , we see that
By (H5), then we get for that
On the other hand, taking and , by (3.8) then and in (). From (3.2) and in () as , we obtain
So we have
Letting , we get
Letting , we get a contradiction. This proves that for some constant C. □
Proof of Theorem 1.5 Under the assumptions (H1)-(H5), we know that the functional Φ given by (2.5) is in . By Lemma 3.2, there exist and such that , where . By Lemma 3.3, for such an r, there exist and suitable such that , where was given before Lemma 3.3. Note that and for , we have
Since from Proposition 2.1 and Remark 1.1 we know that , from (3.2) and Fatou’s lemma, we know that
is and is weakly sequentially lower semicontinuous and is weakly sequentially continuous in . Hence by Proposition 3.1 there exists a (C) c -sequence for Φ, where . By Lemma 3.4, is bounded in E. So, up to a subsequence, we may assume that in E, as . From Lemma 2.2, we know that is compact. So it is easy to check that in . Hence z is a nontrivial solution pair of (1.1). Obviously, is a nonnegative solution pair of (1.1). Applying the strong maximum principle, we obtain that and . This completes the proof. □
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The authors were partially supported by NSFC (No. 11071092; No. 11071095; No. 11101171), the PhD specialized grant of the Ministry of Education of China (20110144110001).
The authors declare that they have no competing interests.
JY is mainly in charge of Section 2 and CW is mainly responsible for Section 3. This paper is finished under of our joint efforts. We discuss many times and make many modifications. Both authors read and approved the final manuscript.