Review

The existence of positive solutions to an elliptic system with nonlinear boundary conditions

Chunhua Wang* and Jing Yang

Author Affiliations

School of Mathematics and Statistics, Huazhong Normal University, Wuhan, 430079, P.R. China

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Boundary Value Problems 2013, 2013:159  doi:10.1186/1687-2770-2013-159

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/159

 Received: 8 April 2013 Accepted: 16 June 2013 Published: 1 July 2013

© 2013 Wang and Yang; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.

Keywords:
fractional Sobolev spaces; linking theorem; nonlinear boundary conditions; strong solution

1 Introduction and main result

In this paper, we mainly study the following system:

(1.1)

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].

We are mainly motivated by [12] and [14].

In [12], Li and one of the authors considered

(1.2)

Under some given conditions, we proved that (1.2) had at least one positive solution pair .

In [14], Bonder, Pinasco, Rossi studied

(1.3)

They assumed that H satisfied the following conditions:

() .

() The Ambrosetti-Rabinowitz type condition: For R large, if ,

(1.4)

where and with

(1.5)

(1.6)

(1.7)

When , they also assumed

(1.8)

() , .

() .

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

(1.9)

(see Lemma 1.1 in [6]). 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 [11]).

The assumptions we impose on and are as follows:

(H1) with for any ,

(H2) uniformly in .

(H3) There is a positive constant such that

(1.10)

where and satisfy

(1.11)

(H4) uniformly in , where , .

(H5) For all , or , , there are two positive constants , such that

(1.12)

where , with for any , .

Remark 1.1 By (1.11), there exist l and m with , such that

(1.13)

Remark 1.2 (H5) was first introduced by Miyagaki and Souto in [15]. 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 [14]. 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 1.5Let (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 [11]) 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 [13].

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

(2.1)

By standard regularity (see [17]), 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 ([18], 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 , ,

(2.2)

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

and by

then satisfies

and hence

Since , we have . Therefore,

Noting that Ω is smooth, it follows from the results of p.187 in [19] (see also [18,20]) that

The following compact result will be useful later.

Proposition 2.1 (Theorem 2.1, [14])

Givenandso that, the inclusion mapis 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 [14]) 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

(2.3)

From (1.10), Remark 1.1 and Proposition 2.1, we can define the functional as

(2.4)

Lemma 2.2The functionaldefined by (2.4) is of classand 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 [21] for the details). □

Now we define the functional for (1.1) given by

(2.5)

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

(2.6)

Now we give a regularity result of an weak solution.

Proposition 2.4Ifis an-weak solution of (1.1), then, and

(2.7)

(2.8)

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 [13], for the readers’ convenience, we give its detailed proof.

Let us consider in (2.6), then

(2.9)

for all .

If we take , then we have

(2.10)

On the other hand, by (1.10) and Proposition 2.1, we have

(2.11)

i.e., . Then from basic elliptic theory (Theorem 9.9, p.9, [17]) there exists one function such that

Then we get

(2.12)

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 [11]. 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

(3.1)

and

(3.2)

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 [11]). 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 [22] that a homotopy , where , is called admissible if:

(i) h is τ-continuous, i.e., in τ-topology as whenever in τ-topology and as .

(ii) 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

and

Proposition 3.1 (Theorem 2.1, [11])

Letbe a separable Hilbert space withorthogonal to. Suppose that

(i) , whereis bounded below, weakly sequentially lower semi-continuous andis weakly sequentially continuous.

(ii) There exist, , andsuch thatand.

Then there exists a (C)c-sequence for Φ, where

Moreover, .

For fixed and , let

Lemma 3.2There existandsuch 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 3.3For thergiven by Lemma 3.2 and anywith, there existssuch that, where.

Proof If , then with either , or , .

(i) If , then we have , and

since for any .

(ii) Assume that . We argue by contradiction. Suppose that there exists a sequence , , , , such that . If , then by the definitions of and , we have

Hence,

Therefore,

Denote , . Then

(3.3)

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

Therefore,

which is impossible.

If , since and as , it follows that . If is such that , we have

thus,

(3.4)

as .

Similarly, if , we have

(3.5)

as .

Since and , we get

Note that

and

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 3.4Ifis a (C)c-sequence of Φ, thenis bounded inE.

Proof Suppose that is a (C)c sequence for Φ, that is,

which shows that

(3.6)

where as .

We suppose, by contradiction, that

(3.7)

and let . Then with

By Proposition 2.1, contains a subsequence, denoted again by such that we may assume that

(3.8)

Let . Then we have

and (3.7) implies that

We may assume, without loss of generality, that

(3.9)

By (H4), we see that

This means that

(3.10)

By (H4), there is an such that

(3.11)

for any and with . Since is continuous on , there is an such that

(3.12)

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

(3.13)

Since by (3.6) we have that

which shows that

(3.14)

Since , we have

(3.15)

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

(3.16)

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. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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.

Acknowledgements

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).

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