Nontrivial Solutions for Asymmetric Kirchhoff Type Problems

and Applied Analysis 3 where μ is an eigenvalue of the problem (19) meaning that there is a nonzero u ∈ E such that


Introduction
We consider the following nonlocal Kirchhoff type problem: where Ω is a smooth bounded domain in ( = 2, 3) and : Ω × → is continuous.
It is pointed out in [1] that the problem (1) models several physical and biological systems where describes a process which depends on the average of itself (e.g., population density). Moreover, this problem is related to the stationary analogue of the Kirchhoff equation which was proposed by Kirchhoff [2] as an extension of the classical D' Alembert wave equation for free vibration of elastic strings. Kirchhoff 's model takes into account the changes in length of the string produced by transverse vibrations. Some early studies of the Kirchhoff equation may be seen [3][4][5]. More recently, by variational methods, Alves et al. [1] and Ma and Rivera [6] studied the existence of one positive solution, and He and Zou [7] studied the existence of infinitely many positive solutions for the problem (1), respectively; Perera and Zhang [8] studied the existence of nontrivial solutions for the problem (1) via the Yang index theory; Zhang and Perera [9] and Mao and Zhang [10] studied the existence of sign-changing solutions for the problem (1) via invariant sets of descent flow. In particular, the asymptotically 4-linear case, lim → 0 ( , ) = , lim → +∞ ( , ) 3 = uniformly in , was considered in [8]. In [9], the authors considered the 4superlinear case: where ( , ) = ∫ 0 ( , ) , which implies that there exists a constant > 0 such that Note that (AR) condition plays an important role for showing the boundedness of Palais-Smale sequences. Furthermore, by a simple calculation, it is easy to see that (AR) condition implies that Hence ( , ) grows in a 4-superlinear rate as | | → +∞.
In the present paper, motivated by [11][12][13][14], our main purpose is to establish existence results of nontrivial solution for the problem (1) with = 2, 3 when the nonlinearity term ( , ⋅) exhibits an asymmetric behavior as ∈ R approaches +∞ and −∞. More precisely, we assume that, for a.e. ∈ Ω, ( , ⋅) grows 4-superlinear at +∞, while at −∞ it has a 4linear growth. To our knowledge, this asymmetric nonlocal Kirchhoff problem is rarely considered by other people. In case of = 3, all the above-mentioned works involve the nonlinear term ( , ) of a subcritical (polynomial) growth; say, (SCP): there exist positive constants 1 and 2 and 0 ∈ (3, 5) such that One of the main reasons to assume this condition (SCP) is that they can use the Sobolev compact embedding 1 0 (Ω) → (Ω), 1 ≤ < 6.
Over the years, many researchers studied the problem (1) by trying to drop the condition (AR); see, for instance, [8,15].
In this paper, our first main results will be to study the problem (1) in the improved subcritical polynomial growth as follows: which is much weaker than (SCP). Note that, in this case, we do not have the Sobolev compact embedding anymore. Our work is studying the asymmetric problem (1) without the (AR) condition in the positive semiaxis. In fact, this condition was studied by Liu and Wang in [16] in the case of Laplacian (i.e., = 2) by the Nehari manifold approach. However, we will use the Mountain Pass Theorem and a suitable version of the Mountain Pass Theorem to get the nontrivial solution to the problem (1) in the case that = 3. Our results are different from those in [8][9][10]15].
We need the following preliminaries. Let := 1 0 (Ω) be the Sobolev space equipped with the inner product and the norm We denote by | ⋅ | the usual -norm. Since Ω is a bounded domain, → (Ω) continuously for ∈ [1,6], compactly for ∈ [1,6), and there exists > 0 such that Recall that function ∈ is called a weak solution of (1) if Seeking a weak solution of the problem (1) is equivalent to finding a critical point * of 1 functional as follows: where ( , ) = ∫ 0 ( , ) . Then there is a subsequence { } such that { } converges strongly in . Also, one says that satisfy the ( ) condition if, for any there is subsequence { } such that { } converges strongly in .
We have the following version of the Mountain Pass Theorem (see [17,18]).

Proposition 2. Let be a real Banach space and suppose that
for some < , > 0, and where Lastly, we also need the following preparations.
Our assumptions lead us to the eigenvalue problem Abstract and Applied Analysis 3 where is an eigenvalue of the problem (19) meaning that there is a nonzero ∈ such that This is called an eigenvector corresponding to eigenvalue . Set Denote by 0 < 1 < 2 < ⋅ ⋅ ⋅ all distinct eigenvalues of the nonlinear problem (19). Then where 1 > 0 is simple and isolated and 1 can be achieved at some 1 ∈ and 1 > 0 in Ω (see [9]).
Theorem 4. Let = 3 and assume that has the improved subcritical polynomial growth on Ω (condition (SCPI)) and uniformly, for a.e. ∈ Ω, then the problem (1) has at least one nontrivial solution.
In case of = 2, we have 2 * = +∞. In this case, every polynomial growth is admitted, but one knows easy examples that ̸ ⊆ ∞ (Ω). Hence, one is led to look for a function ( ) : R → + with maximal growth such that It was shown by Trudinger [19] and Moser [20] that the maximal growth is of exponential type. So we must redefine the subcritical (exponential) growth in this case as follows.

Abstract and Applied Analysis
On the other hand, if ∈ ( 1 , +∞), taking > 0 such that − > 1 and using (26), we have Thus part (ii) is proved. By exactly slight modification to the proof above, we can prove (ii) if = +∞.
On the other hand, if ∈ ( 1 , +∞), taking > 0 such that − > 1 and using (31), we have Thus part (ii) is proved. By exactly slight modification to the proof above, we can prove (ii) if = +∞. By exactly slight modification to the proof above, we can prove (ii) if = +∞.

Proofs of the Main Results
Proof of Theorem 3. By Lemma 9, the geometry conditions of Mountain Mass Theorem hold. So we only need to verify condition (PS). Let { } ⊂ be a (PS) sequence such that, for every ∈ N, where > 0 is a positive constant and { } ⊂ R + is a sequence which converges to zero.
Step 2. Now, we prove that { } has a convergence subsequence. In fact, we can suppose that Now, since has the subcritical growth on Ω, for every > 0, we can find a constant ( ) > 0 such that Then Similarly, since ⇀ in , ∫ Ω | − | → 0. Since > 0 is arbitrary, we can conclude that From (62) and (63), we obtain → ‖ ‖ as → ∞.
So we have → in which means that satisfies (PS).
Proof of Theorem 4. Since = 1 , obviously, Lemma 9 (i) holds. We only need to show that Lemma 9 (ii) holds. Let = 1 . Using the condition ( 3 ), then there exists > 0 large enough such that for all ∈ Ω and large enough. So we have To complete our proof, we first need to verify that { } is bounded in . Similar to the proof of Theorem 3, we have 0 ( ) ≤ 0, ∈ Ω, 0 ( ) ̸ ≡ 0, and for all V ∈ . By maximum principle, 0 < 0 is an eigenfunction of 1 ; then | ( )| → ∞ for a.e. ∈ Ω. By our assumptions, we have which contradicts (72). Hence { } is bounded. According to the Step 2 proof of Theorem 3, we have → in which means that satisfies ( ).