Existence of solutions of elliptic boundary value problems with mixed type nonlinearities
© Mao et al.; licensee Springer 2012
Received: 14 May 2012
Accepted: 20 August 2012
Published: 31 August 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.
where () is a bounded open domain with a smooth boundary ∂ Ω and .
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. .
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.
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.
() 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;
A straightforward computation shows that K and satisfy the assumptions of Theorem 1.1, but neither nor satisfy the (AR) condition.
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
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 )
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.
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 .
Lemma 3.2 Under the assumptions of Theorem 1.1, there exists such that .
Lemma 3.3 Under the assumptions of Theorem 1.1, the functional Φ satisfies the (C) condition.
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 )
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 ,
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. □
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.
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