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Existence of solutions of elliptic boundary value problems with mixed type nonlinearities
Boundary Value Problemsvolume 2012, Article number: 97 (2012)
We study the existence of a nontrivial solution of the following elliptic boundary value problem with mixed type nonlinearities:
where . We consider the problem in a different case: , is some constant. Assuming that K satisfies the “pinching” condition, and W satisfies a more general superquadratic growth condition than the well-known Ambrosetti-Rabinowitz condition usually used in literature, we obtain a nontrivial solution via the Mountain Pass Lemma.
MSC:35J65, 35J20, 47J10.
In this paper, we shall be concerned with the elliptic boundary value problem in a different case
where () is a bounded open domain with a smooth boundary ∂ Ω and .
The existence of nontrivial weak solutions for (P) have been studied in many papers, see [1–12]. Su and Zhao in  considered problem (P) for resonance case at infinity, , where is an eigenvalue of the linear boundary value problem
the existence of multiple nontrivial solutions for (P) are obtained by minimax methods and Morse theory. Ambrosetti and Rabinowitz in  established the existence of a nontrivial solution for problem (P) by assuming the following conditions:
() , , uniformly in a.e. .
() There exist two positive constants a and b such that
And the following well-known Ambrosetti-Rabinowitz condition ((AR) for short):
Since then, the (AR) condition has been used extensively in many literature sources, see [12–18]. It is well known that the (AR) condition is quite natural and convenient not only to ensure that the Euler-Lagrange functional associated to problem (P) has a mountain pass geometry but also to guarantee that the Palais-Smale sequence of the Euler-Lagrange functional is bounded. Let E be a Hilbert space and . Recall that the sequence is said to be a Palais-Smale sequence of G provided that is bounded and as , the function G satisfies the Palais-Smale condition ((PS) for short) if and only if any Palais-Smale sequence for G contains a convergent subsequence. The function G satisfies the Cerami condition ((C) for short) if any sequence in E satisfying is bounded and as has a convergent subsequence.
Without (AR), it becomes more complicated. Indeed, there are many functions which are superlinear, but it is not necessary to satisfy (AR) even if . Willem and Zou stated the following examples:
where . Then it is easy to check that (AR) does not hold even for any . On the other hand, in order to verify (AR), it usually is an annoying task to compute a primitive function of f and sometimes it is almost impossible. For example,
Some authors have tried to drop or weaken the above superlinear condition (AR) in recent years, see [4–8, 11, 19]. Miyagaki and Souto  adapted some monotonicity arguments studying the existence of nontrivial weak solutions of (P).
The aim of the manuscript is to consider the problem in a different case: , is some constant. We study this problem under “pinching” condition and the general superquadratic condition. The case that has a part with “pinching” condition has been considered only by few authors, see [13, 20]. Since does not satisfy the () and (AR), problem (P) becomes more delicate. The main difficulty when dealing with this problem is the lack of compactness of the Sobolev embedding theorem.
In this paper, here, replaced by , satisfy
() , are -maps.
() There are two positive constants and such that
() There exists such that
() and as uniformly in x.
() Set , if , as uniformly in x, and there exist and such that if .
We will prove the following results.
Theorem 1.1 If assumptions (), (), () and ()-() are satisfied, then problem (P) has a nontrivial weak solution.
Our assumptions (), () are weaker than (AR), and there is no monotone condition;
Example 1 Consider the functions
A straightforward computation shows that K and satisfy the assumptions of Theorem 1.1, but neither nor satisfy the (AR) condition.
Example 2 Consider the more general functions
where is of superlinear growth as . A straightforward computation shows that K and W satisfy the assumptions of Theorem 1.1.
We will prove that the function associated with (P) has Mountain Pass geometry and satisfies the condition. The remainder of the paper is organized as follows. In Section 2, we deal with the variational setting. In Section 3, we give the details of the proof of Theorem 1.1.
2 Preliminary results
Let be the Sobolev space equipped with the inner product and the norm
And we denote the usual -norm
Our approach will be the variational techniques. Define the Euler-Lagrange functional associated to problem (P) given by
From the assumptions on f, it is standard to check that whose Gateaux derivative is
Let be given by
By () and set , ,
It is worth pointing out that if the function is of the form with and then η in a Hilbert space is equivalent to the norm ; however, if the function is not of the form , η is not a norm because of the lack of norm’s linear property.
Lemma 2.1 (see )
Let H be a real Banach space, , satisfying . Moreover,
there exist such that ,
there exists such that .
Then there exists a sequence such that and as .
Lemma 2.2 (see )
Assume that , , , and . Then, for every , and the operator , is continuous.
3 Proofs of theorems
First of all, we recall a property of the function , which is necessary to the proof of the geometric structure of the functional Φ.
Fact 1 Assume that () holds, then
Proof Define ,
By (), , which implies is non-decreasing. So, we have
Next we discuss the geometric structure of the functional Φ on H. □
Lemma 3.1 Under the assumptions of Theorem 1.1, there are constants such that .
Proof From () and (), as , we have
where is a positive constant. Hence as ,
From , we know , so we can choose . And using () again, we observe that for any given there is such that
It follows from (3.2) and the Sobolev embedding theorem that for all
where c is a positive constant. Then combining () and (3.3), we obtain
set , it is clear that . We choose and , then
Lemma 3.2 Under the assumptions of Theorem 1.1, there exists such that .
Proof Let , and . By (), there exists such that
As , by Fact 1, we have
Then, by inequalities (3.4) and (3.5), we get
By the choice of A, we have , so there exists such that if , then
Suppose that the assumptions of Theorem 1.1 hold, we have Lemma 3.1 and Lemma 3.2. Now it follows from Lemma 2.1 that there is a sequence such that
Lemma 3.3 Under the assumptions of Theorem 1.1, the functional Φ satisfies the (C) condition.
Proof Let be such that
By () we observe that for large n,
Arguing indirectly, assume as a contradiction that . Setting , then and since the embedding for , we have . Observe that, from (3.8), () and ()
It follows that for any and n large enough,
By (), for all and as . For , let
Since if , one has and
It follows from (3.9) that
Set , since , one sees . Fix arbitrarily , using (3.11),
which implies by the Hölder inequality that
as uniformly in n. Using (3.11) again, for any fixed ,
as . Let , by (), there exists such that
for all n. By () and (3.12), we can take large so that
Hence combining (3.11), (3.12) and (3.14), there is such that
for . Note that there is independent of n such that
for all . Therefore, combining (3.14)-(3.16), we obtain for
which contradicts (3.10). Hence is bounded in H. Going if necessary to a subsequence, we assume that
which implies a.e. in Ω, because the imbedding is compact. Hence we have and . Using the Hölder inequality
() for in , and by (), (), (), () we have
Then, by Lemma 2.2, we have in . Thus
as . Moreover, a straightforward computation shows that
it is clear that
This completes the proof. □
Now, we are ready to prove Theorem 1.1.
We will obtain a critical point of by the use of a standard version of the Mountain Pass Lemma (see ). It provides the minimax characterization for the critical value which is important for what follows. Therefore, we state this lemma precisely.
Lemma 3.4 (see )
Let H be a real Banach space and be a -smooth functional. If Φ satisfies the following conditions:
every sequence in H such that is bounded in and in as , contains a convergent subsequence ((PS) condition),
there are constants such that ,
there is a constant such that ,
where is an open ball in H of radius ρ centered at 0, then Φ possesses a critical value given by
Now we are ready to give the proofs of Theorem 1.1.
Proof of Theorem 1.1 Under conditions (), (), (), ()-(), as shown in , a deformation lemma can be proved with the (C) condition, replacing the usual Palais-Smale condition, and it turns out that the Mountain Pass Theorem still holds true. Applying the Mountain Pass Lemma 3.4, Φ possesses a critical value given by . Hence, u is a nontrivial solution of problem (P) satisfying , . The proof is done. □
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The authors are grateful to anonymous referees for detailed reading of the manuscript and valuable comments, which helped us improve this work. This work was supported by The National Natural Science Foundation of China (No. 11101237) and SNSFC ZR2012AM006.
The authors declare that they have no competing interests.
The paper is the result of joint work of all authors who contributed equally to the final version of the paper. All authors read and approved the final manuscript.