Positive Solutions for a Nonhomogeneous Kirchhoff Equation with the Asymptotical Nonlinearity in R 3

and Applied Analysis 3 then (k1) holds. To verify the condition (k2), we have to choose some special R 0 > 0. For any R > 0, taking ψ ∈ C ∞ 0 (R 3 , [0, 1]) such that ψ(x) = 1 if |x| ≤ R, ψ(x) = 0 if |x| ≥ 2R and |∇ψ(x)| ≤ C/√aR for all x ∈ R, where C > 0 is an arbitrary constant independent of x. Then, for any R 0 > 2R, we have ∫ R 3 k (x) F (ψ) dx ≥ ∫ |x|≤R k (x) F (ψ) dx


Introduction and Main Results
In this paper, we consider the following nonhomogeneous Kirchhoff equation: where constants ,  > 0, and functions , and ℎ satisfy the following conditions:  is a positive bounded condition,  ∈ (,  + ), () ≡ 0 if  < 0 and ℎ ∈  2 ( 3 ), ℎ ≥ 0. Note that, with  = 1,  = 0, and  3 replaced by   , problem (1) reduces to which can be looked at as a generalization of the well known Schrödinger equation.
When Ω 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 [1]) 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, ℎ is the area of the cross section,  is the Young modulus of the material,  is the mass density, and  0 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][3][4][5][6][7][8] and the references therein.Some interesting studies for Kirchhoff-type problem (3) in a bounded domain Ω of   by variational methods can be found in [2,[9][10][11][12][13][14][15][16][17][18][19][20][21][22].Very recently, some authors had studied the Kirchhoff equation on the whole space   and obtained the existence of multiple solutions (see [23][24][25][26][27][28][29][30][31]).In the same spirit of [24][25][26][28][29][30][31], we study a nonhomogeneous Kirchhoff equation (1) on the whole space  3 .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 2 Abstract and Applied Analysis [23] study it under the condition of superlinear nonlinearity at infinity.In [33], Wang and Zhou study the existence of two positive solutions for a nonhomogeneous elliptic equation ((1) with  = 1 and  = 0).In [32], 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 ,  > 0, the asymptotically linear nonlinearity at infinity and the nonhomogeneous term. S, we can not obtain the existence of a ground state solution for Kirchhoff-type problem (1) and the compactness result as in [32] because of the nonhomogeneous term, and we cannot easily obtain the compactness result as in [33] due to the nonlocal term (or  ̸ = 0).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 1 ≤  ≤ +∞, we denote by ‖ ⋅ ‖  the usual norm of the Lebesgue space   ( 3 ).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  :  1 ( 3 ) →  of ( 1) is defined by By Lemma 2.1 in [24] or Lemma 1 in [25], the functional  is  1 ( 1 ( 3 ), ) with the derivative given by Hence, if  ∈  1 ( 3 ) is a nonzero critical point of , then it is also a nonnegative solution of (1).In fact, by () ≡ 0 if  < 0 and ℎ ≥ 0, we have , where  − = max{−, 0}.This yields that  − = 0; then  =  + −  − =  + ≥ 0, where  + = max{, 0}.By the maximum principle, the nonzero critical point of  is the positive solution for problem (1).
Here is the main result of this paper.
Remark 3. If ℎ ≡ 0, we know that problem (1) has a positive ground state solution by using the method in [32] and a trivial solution (() ≡ 0).If ℎ ̸ ≡ 0, a trivial solution (() ≡ 0) 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 ‖ℎ‖ 2 <  for small .
In order to obtain our results, we have to overcome various difficulties.Since the embedding of  1 ( 3 ) into   ( 3 ),  ∈ [2,6], 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 ∫ ||≥ (|∇  | 2 + |  | 2 ) ≤  which is similar to [32] but different from the one in [24][25][26][28][29][30][31].In fact, this difficulty can be avoided, when problems are considered, restricting  to the subspace of  1 ( 3 ) consisting of radially symmetric functions [23,24,29] and constraint potential functions [25,30], or when one is looking for semiclassical states [28], 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   ( = 1, 2, 3, . ..) for convenience.

Proof of Main Result
In this section, we prove that problem (1) has a mountain pass type solution and a local minimum solution with ℎ ̸ ≡ 0. For this purpose, we use a variant version of Mountain Pass Theorem [35], 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 ℎ ∈  2 ( 3 ), ℎ ≥ 0, (f1)-(f3), and (k1) hold.Then there exist , ,  > 0 such that ()| ‖‖= ≥  > 0 for ‖ℎ‖ 2 < .
) and there is a  ∈  1 ( 3 ) such that, up to a sequence, as  → ∞.