1 Introduction and main result
In this paper, we study the following Kirchhoff type equations in ():
where , and f satisfy the following assumptions:
, if and as .
There exists such that as .
is a continuous function and there exists such that
Throughout this paper, we denote by the usual Sobolev space equipped with the following inner product and norm:
Define the energy functional by
where . The functional is well defined for each and belongs to . Moreover, for any , we have
Clearly, the critical points of are the weak solutions for problem (1.1).
In recent years, much attention has been paid to Kirchhoff type equations. Let be a bounded domain. The following Kirchhoff problem with zero boundary data:
which is related to the stationary analogue of the equation
was proposed by Kirchhoff  as an extension of the classical D’Alembert’s wave equation for free vibrations of elastic strings. Kirchhoff’s model takes into account the changes in length of the string produced by transverse vibrations. In (1.2), u denotes the displacement, the external force and b the initial tension, while a is related to the intrinsic properties of the string, such as Young’s modulus. We would like to point out that such nonlocal problems also appear in other fields such as biological systems, where u describes a process which depends on the average of itself, for example, population density. It is worth mentioning that Fiscella and Valdinoci  proposed a stationary fractional Kirchhoff model, in bounded regular domains of , which takes into account the nonlocal aspect of the tension arising from nonlocal measurements of the fractional length of the string. For some recent results about stationary Kirchhoff problems involving the fractional Laplacian, we refer to [18, 20, 19, 22, 23, 25, 30, 31, 34] and the references therein. For more mathematical and physical background for problem (1.2), we refer the readers to [1, 2, 5, 24] and the references therein.
Recently, problems like type (1.2) in bounded domains have been investigated by many authors, see, for instance, [6, 7, 16, 17, 21, 26, 32, 33]. In , Ma and Muñoz Rivera obtained positive solutions via variational methods. In , Perera and Zhang obtained a nontrivial solution via the Yang index and the critical group. Zhang and Perera , and Mao and Zhang  obtained multiple and sign-changing solutions via the invariant sets of descent flow. Shuai  obtained one least energy sign-changing solution via a constraint variational method and the quantitative deformation lemma. He and Zou [6, 7] obtained infinitely many solutions via the local minimum method and the fountain theorems.
Equations of type (1.1) in the whole space (), but with the nonlinear term being replaced by a more general nonlinear term , have also been studied extensively, see, for example, [8, 9, 15, 13, 12, 28, 35] and the references therein. More recently, the researchers paid their attention on asymptotically linear Kirchhoff equations. In , for a special type of a Kirchhoff equation with asymptotically linear term, in which the nonlocal term is like (this makes the functional contain a term like ), Li and Sun only needed to verify the (PS) condition in order to apply the mountain pass theorem to obtain the existence and multiplicity of solutions. Moreover, we noticed that the potential in  was assumed to be radially symmetric, in order to consider the problem in a radial function Sobolev space in which the compact Sobolev embedding holds, and thus being easy to verify the (PS) condition. In some cases, this symmetric condition may be replaced by other compact conditions which make the compact embedding holds; here we just quote . In , Wu and Liu studied the existence and multiplicity of nontrivial solutions for a Kirchhoff equation in with asymptotically linear term via Morse theory and local linking. To overcome the loss of compactness, the usual strategy is to restrict the functional to a subspace of , which embeds compactly into with certain assumptions on radially symmetric functions. In this paper, we will study directly problem (1.1) in H, not in any subspaces. To overcome the loss of compactness, motivated by  in which Liu, Wang and Zhou studied an asymptotically linear Schrödinger equation, we will make a careful prior estimate for the Cerami sequence (defined later). Unlike the problem in , we not only show the convergence of the Cerami sequence, but also show that as , where is a Cerami sequence. This makes the study of our problem more difficult. Through a careful observation, we show that this limit holds only by assuming that (A1), (A2) and (A3) hold, without any more assumptions.
Now let us state the main result of this paper.
then there exists such that for any , problem (1.1) admits at least two nontrivial nonnegative solutions, in which one is a mountain pass type solution and the other is a ground state solution.
We use the following notation:
C, , , etc. will denote positive constants whose exact values are not essential.
is the duality pairing between and H, where denotes the dual space of H.
is the norm of the space .
denotes the open ball centered at x having radius R.
2 Preliminary lemmas
To prove Theorem 1.1, we use a variant version of the mountain pass theorem, which allows us to find a so-called Cerami type (PS) sequence. The properties of this kind of (PS) sequence are very helpful in showing the boundedness of the sequence in the asymptotically linear case.
Theorem 2.1 ().
Let E be a real Banach space with its dual space , and suppose that satisfies
for some , and with . Let be characterized by
where is the set of continuous paths joining 0 and e. Then there exists a sequence such that
This kind of sequence is usually called a Cerami sequence.
Thus, one has
thanks to the fact that is nonnegative. Fixing and letting be small enough, there exists such that the desired conclusion holds. ∎
According to the definition of μ and , there exists , with , such that
By (A2) and Fatou’s Lemma, we have
by taking with large enough so that and . Since
is continuous and increasing in and , there exists sufficiently small such that for all . ∎
In order to get the existence of a nontrivial nonnegative solution, we first show that this sequence is bounded.
Assume on the contrary that as . Define . Clearly, is bounded in H and there exists such that, going if necessary to a subsequence,
We claim that . Assume on the contrary that . By (A3), there exists a constant such that
For any , this yields
Since the embedding is compact, we have in . According to [27, Lemma A.1], going if necessary to a subsequence, there exists such that
Noting that a.e. in , we obtain
Since as , it follows from (2.4) that
which contradicts (2.12), so .
On the other hand, since as , it follows form (2.4) that
Fatou’s Lemma yields
Since the embedding is continuous, ω also belongs to . According to the definition of the norm of and (2.15), . This is a contradiction. So, the sequence is bounded in H. ∎
To prove that the Cerami sequence in (2.4) converges to a nonzero critical point of , we need the following compactness lemma.
for all and .
Let be a smooth function such that
and, for some constant (independent of R),
Then, for all and , we have
This implies that
for all and . By (2.4), as . So, for any , there exists such that
for all and . Note that
For any , there exists such that
By (2.21) and Young’s inequality, for all and , we get
From and (2.16), the desired conclusion easily follows. ∎
3 Proof of Theorem 1.1
Now we are in a position to give the proof of Theorem 1.1.
Proof of Theorem 1.1.
By Lemma 2.3, the sequence defined in (2.4) is bounded in H. Since H is a reflexive space, going if necessary to a subsequence, in H for some . In order to prove the theorem, we need to show that and that the sequence has a strong convergence subsequence in H, that is, as . Note that, by (2.4),
Since in H, we have
Since the embedding is continuous, in , and thus
To show that
we first prove that
This and the compactness of the embedding imply (3.5). Since is bounded in , we assume that . If , by Fatou’s Lemma,
By Fatou’s Lemma, it is easy to see from our assumption that
which is impossible. So,
Moreover, we get as . Now we show that the solution u is nonnegative. Multiplying equation (1.1) by and integrating over , where , we find
Hence, and u is a nonnegative solution of problem (1.1).
To get a ground state solution, we denote by K the nontrivial critical set of . Set
It is easy to see that K is nonempty. For any , we have
Therefore, for any , we have
We recall that whenever , and (3.8) implies
Hence, any limit point of a sequence in K is different from zero.
We claim that is bounded from below on K, i.e., there exists such that for all . Otherwise, there exists such that
It follows from (2.3) that
This and (3.10) imply that as . Let . There exists such that (2.5) holds. Note that for . As in the proof of Lemma 2.3, we obtain that is impossible. Then is bounded from below on K. So . Let be such that as . Then (2.4) holds for the sequence and m. Following almost the same procedures as in the proofs of Lemmas 2.3 and 2.4, and using the above arguments, we can show that is bounded in H and, going if necessary to a subsequence, , where and as . There is only one difference in showing that (3.6) does not hold. Based on the aforementioned discussions, we know that the possible critical value . Here we do not know if , but (3.9) holds, and so . Moreover, and . Therefore, is a ground state solution of problem (1.1). Finally, we can also show that the ground state solution is nonnegative. ∎
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About the article
Published Online: 2017-03-16
Funding Source: National Natural Science Foundation of China
Award identifier / Grant number: 11301181
Funding Source: Natural Science Foundation of Heilongjiang Province
Award identifier / Grant number: A201306
The first author is supported by NSFC (No. 11301181) and China Postdoctoral Science Foundation, the second author is supported by Young Teachers Foundation of BTBU (No. QNJJ2016-15). The third author was supported by Natural Science Foundation of Heilongjiang Province of China (No. A201306) and Research Foundation of Heilongjiang Educational Committee (No. 12541667) and Doctoral Research Foundation of Heilongjiang Institute of Technology (No. 2013BJ15).
Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 267–277, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2016-0240.
© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0