Research Article | Open Access
Positive Solutions for a Nonhomogeneous Kirchhoff Equation with the Asymptotical Nonlinearity in
We study the following nonhomogeneous Kirchhoff equation: , where is asymptotically linear with respect to at infinity. Under appropriate assumptions on , and , existence of two positive solutions is proved by using the Ekeland's variational principle and the Mountain Pass Theorem in critical point theory.
1. Introduction and Main Results
In this paper, we consider the following nonhomogeneous Kirchhoff equation: where constants , and functions , and satisfy the following conditions: is a positive bounded condition, , if and , . Note that, with , , and replaced by , problem (1) reduces to which can be looked at as a generalization of the well known Schrödinger equation.
When is a smooth bounded domain in , the problem is related to the stationary analogue of the Kirchhoff equation which was proposed by Kirchhoff in 1883 (see ) as a generalization of the well known 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. Here, is the length of the string, is the area of the cross section, is the Young modulus of the material, is the mass density, and is the initial tension. Moreover, Kirchhoff’s type problems also model several physical systems and biological systems and there are many interesting results for problem (3) which can be found in [2–8] and the references therein.
Some interesting studies for Kirchhoff-type problem (3) in a bounded domain of by variational methods can be found in [2, 9–22]. Very recently, some authors had studied the Kirchhoff equation on the whole space and obtained the existence of multiple solutions (see [23–31]). In the same spirit of [24–26, 28–31], we study a nonhomogeneous Kirchhoff equation (1) on the whole space . Especially, inspired by the paper [32, 33], we consider the asymptotically linear nonlinearity at infinity of problem (1). For the nonhomogeneous Kirchhoff problem, Chen and Li in  study it under the condition of superlinear nonlinearity at infinity. In , Wang and Zhou study the existence of two positive solutions for a nonhomogeneous elliptic equation ((1) with and ). In , Sun et al. study the existence of a ground state solution for some nonautonomous Schrödinger-Poisson systems involving the asymptotically linear nonlinearity at infinity without the nonhomogeneous term. But we will study the existence of two positive solutions for Kirchhoff-type problem (1) with , the asymptotically linear nonlinearity at infinity and the nonhomogeneous term. So, we can not obtain the existence of a ground state solution for Kirchhoff-type problem (1) and the compactness result as in  because of the nonhomogeneous term, and we cannot easily obtain the compactness result as in  due to the nonlocal term (or ). To our best knowledge, little has been done for nonhomogeneous Kirchhoff problems with respect to the asymptotically linear nonlinearity at infinity.
Before stating our main results, we give some notations. For any , we denote by the usual norm of the Lebesgue space . Define the function space with the product and equivalent norm Define the function space with the standard product and norm Recall that the Sobolev’s inequality with the best constant is Moreover, problem (1) has a variational structure. Indeed the corresponding action functional of (1) is defined by By Lemma 2.1 in  or Lemma 1 in , the functional is with the derivative given by Hence, if is a nonzero critical point of , then it is also a nonnegative solution of (1). In fact, by if and , we have , where . This yields that ; then , where . By the maximum principle, the nonzero critical point of is the positive solution for problem (1).
Here is the main result of this paper.
Theorem 1. Suppose that , , and the following conditions hold.(f1), , and for .(f2). (f3). (k1) is a positive continuous function and there exists such that
hold, where , .
Then problem (1) has at least two positive solutions satisfying and if for some small .
Remark 2. It is not difficult to find some functions satisfying conditions of Theorem 1. For example, for any , let Clearly, satisfies (f1)–(f3) with . Moreover, and . Taking a positive continuous function , where for some . Note that then (k1) holds. To verify the condition (k2), we have to choose some special . For any , taking such that if , if and for all , where is an arbitrary constant independent of . Then, for any , we have Taking , , where is large enough such that , , and . Let . Then, we obtain that . Moreover, in view of the definition of and (18), one has So, condition (k2) holds. In particular, the condition (k1) and above examples can also be found in  in which the asymptotically linear term satisfying (k1) appeared first.
Remark 3. If , we know that problem (1) has a positive ground state solution by using the method in  and a trivial solution (). If , a trivial solution () is replaced by the local minimum solution by Theorem 1. Note that the local minimal solution exists due to the homogeneous term which is looked at as a small perturbation because for small .
In order to obtain our results, we have to overcome various difficulties. Since the embedding of into , , is not compact, condition (k1) and (k2) are crucial to obtain the boundedness of Cerami sequence. Furthermore, in order to recover the compactness, we establish a compactness result which is similar to  but different from the one in [24–26, 28–31]. In fact, this difficulty can be avoided, when problems are considered, restricting to the subspace of consisting of radially symmetric functions [23, 24, 29] and constraint potential functions [25, 30], or when one is looking for semiclassical states , by using perturbation methods or a reduction to a finite dimension by the projections method. Third, it is not difficult to find that every (PS) sequence is bounded because a variant of Ambrosetti-Rabinowitz condition is satisfied (see [23, 25, 31]). However, for the asymptotically linear case, we have to find another method to verify the boundedness of (PS) sequence.
This paper is organized as follows. In Section 2, we manage to give proofs of Theorem 1. In the following discussion, we denote various positive constants as or for convenience.
2. Proof of Main Result
In this section, we prove that problem (1) has a mountain pass type solution and a local minimum solution with . For this purpose, we use a variant version of Mountain Pass Theorem , which allows us to find a so-called Cerami type PS sequence (Cerami sequence, in short). The properties of this kind of Cerami sequence sequences are very helpful in showing its boundedness in the asymptotical cases. The following lemmas will show that has the so-called mountain pass geometry.
Lemma 4. Suppose that , , (f1)–(f3), and (k1) hold. Then there exist such that for .
Proof. For any , it follows from (f1)–(f3) that there exists such that Therefore, we have Furthermore, by (f1)–(f3) and (k1), there exists such that According to (21), (22), and the Sobolev inequality, we deduce that where . By , , and the Hlder inequality, one has Taking and setting for , we see there exists such that . Then it follows from (24) that there exists such that for . Of course, can be chosen small enough; we can obtain the same result: there exists such that for .
Lemma 5. Suppose that , , (f1)–(f3), and (k1)-(k2) hold. Then there exists with , is given by Lemma 4, such that .
Proof. By (k2) and , in view of the definition of and with , there is a nonnegative function such that
and . Then, we have
Choosing small enough in Lemma 4 such that , then this Lemma is proved.
From Lemmas 4 and 5 and Mountain Pass Lemma in , there is a Cerami sequence such that where denotes the dual space of . In the following Lemmas 6 and 7, we shall prove that satisfies the Cerami condition, that is; the Cerami sequence has a convergence subsequence.
Lemma 6. Suppose that , , (f1)–(f3), and (k1) hold. Then defined in (27) is bounded in .
Proof. By contradiction, let . Define . Clearly, is bounded in and there is a such that, up to a sequence,
Firstly, we claim that is nontrivial; that is, . Otherwise, if , the Sobolev embedding implies that strongly in ; is given by (k1). By (f1)–(f3), there exists such that Then, for all , we have This yields Furthermore, by (k1), there exists a constant such that Then, for all , we have Combining (31) and (33), we obtain By (27), we get as . Together with as , it follows that Together with , we have where, and in what follows, denotes a quantity which goes to zero as . Therefore, we deduce that which contradicts (34). So, .
Furthermore, because as , it follows from (35) that that is, Together with (22), (29), and , one has This yields This means Therefore, we have By , we get as ; thus strongly in ; therefore, weakly in . Since weakly in ; we have weakly in . By the uniqueness of the weak limitation, we have which contradicts . Therefore, the Cerami sequence is bounded in .
Lemma 7. Suppose that , , (f1)–(f3), and (k1) hold. Then for any , there exist and such that defined in (27) satisfies for and .
Proof. Let be a smooth function such that Moreover, there exists a constant independent of such that Then, for all and , by (45), (46), and the Hlder inequality, we have This implies that for all and , where . From Lemma 6, we know that is bounded in . Together with (27), we obtain that in . Moreover, by (48), for , there exists such that for and . Note that This yields By (32), we have This yields for all and . For any , there exists such that Because , , there exists such that By the Hlder inequality, (45), (55), and the boundedness of in , we have By the Young inequality, (46), and (54), for all and , we obtain Combining , (51), (53), (56), (57), and the boundedness of in , there exists such that Note that is independent of . So, for any , we can choose and such that holds.
Lemma 8. Suppose that , , (f1)–(f3), and (k1)-(k2) hold. Then the sequence in (27) has a convergent subsequence. Moreover, possesses a nonzero critical point in and .
Proof. By Lemma 6, the sequence in (27) is bounded in . We may assume that up to a subsequence weakly in for some . Now, we shall show that as .
By (11), we have By (59), , and the boundedness of in , we easily get One has It is clear that
Since weakly in , we obtain By the continuity of imbedding , we have that weakly in ; that is, By (63) and (64), we deduce Combining the boundedness of in , (64), and (65), we obtain
Moreover, by (32), Lemma 7, and in , for any and large enough, one has This and the compactness of embedding imply that
Since is bounded in and the continuity of the Sobolev embedding of imbedding in , for any choice of and , the relation holds for large . By , for any there exists such that By (70) and (69), we have This yields By (61), (62), (66), (68), and (72), we have