- Open Access
The existence of positive solutions to an elliptic system with nonlinear boundary conditions
© Wang and Yang; licensee Springer 2013
- Received: 8 April 2013
- Accepted: 16 June 2013
- Published: 1 July 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.
- fractional Sobolev spaces
- linking theorem
- nonlinear boundary conditions
- strong solution
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].
Under some given conditions, we proved that (1.2) had at least one positive solution pair .
They assumed that H satisfied the following conditions:
() , .
(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:
(H2) uniformly in .
(H4) uniformly in , where , .
where , with for any , .
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.
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.
In this section we mainly give some preliminaries which will be used in Section 3. We follow the structure in .
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.
Moreover, is compact.
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). □
Moreover, Φ is class .
Now we give a regularity result of an weak solution.
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.
for all .
which implies that . We have gotten that . Finally, since , we conclude that u satisfies (2.7). We can make the same argument for v. □
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.
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]).
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 .
Proposition 3.1 (Theorem 2.1, )
, where is bounded below, weakly sequentially lower semi-continuous and is weakly sequentially continuous.
There exist , , and such that and .
Lemma 3.2 There exist and such that .
for some . □
Lemma 3.3 For the r given by Lemma 3.2 and any with , there exists such that , where .
- (i)If , then we have , and
- (ii)Assume that . We argue by contradiction. Suppose that there exists a sequence , , , , such that . If , then by the definitions of and , we have
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.
which is impossible.
which is impossible, thus the lemma is proved. □
Lemma 3.4 If is a (C) c -sequence of Φ, then is bounded in E.
where as .
We claim that .
which is impossible.
Hence a.e. in ∂ Ω.
Letting , we get a contradiction. This proves that for some constant C. □
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. □
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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