The Existence of Multiple Solutions for Nonhomogeneous Kirchhoff Type Equations in R 3

and Applied Analysis 3 since p > 1, by calculating directly, we see that max t⩾0 g(t) = g(ρ) > 0, where ρ = ((p + 1)/2pγp+1 p+1 ) 1/(p−1), g(ρ) = ((p − 1)/2p)ρ. Then it follows that, if |h| 2 γ 2 < g(ρ), that is, |h| 2 < γ −1 2 g(ρ) ≜ m p , there exists α = ρ(g(ρ) − γ 2 |h| 2 ) > 0 such that J| ∂Bρ ⩾ α, where

Under the above assumptions, the mountain pass geometry structure and the boundedness of Palais-Smale sequence or Cerami sequence can be obtained.
For example, in [5], when  satisfies above assumptions and the potential  satisfies the following conditions: (V)  ∈ (R  , R), inf R   > 0 and for each  > 0, meas{ ∈ R  : () ⩽ } < ∞, where meas denotes the Lebesgue measure, which ensure the compact imbedding of ), the author obtained the existence of a nontrivial solution to problem (3).The existence of infinitely many solutions was considered in [2,3] respectively, by the fountain theorem and a variant version of the fountain theorem, where  is odd on  ∈ R and is also subcritical, superlinear at the origin, and either 4-superlinear at infinity or satisfies AR condition or some conditions weaker than AR condition.In [2],  = 2, 3,  ≡ 1 and in [3],  = 3,  ∈  ∞ loc (R 3 ) satisfies the condition (V).The existence of ground state solutions to problem (3) was also considered in [1,4].In [1], the authors studied (3) under the conditions:  = 3, a positive potential satisfies (3,5) and ()/ 3 increases for all  > 0. They obtained a positive ground state solution by using the Nehari manifold.
Under the same condition of  in [1], the authors in [4] discussed the existence of multiple ground state solutions, where (, ) = () + || 4 , which contains a critical growth term.
Recently, in [6], the authors studied the existence of a positive solution for the following Kirchhoff equation: where  ⩾ 3, ,  > 0,  is subcritical, superlinear at the origin and infinity.In order to construct the mountain pass geometry structure and obtain the bounded PS sequence, they combined a truncation argument with a monotonicity trick introduced by Jeanjean [7], and obtained that there exists  0 > 0 such that problem (4) has at least one positive solution for  ∈ (0,  0 ).
Motivated by the aformentioned references, we consider the existence of multiple solutions to the nonhomogeneous Kirchhoff equation (1), where  ∈ (1,5).By using the variational method, we obtain that the problem has at least two positive radial solutions.Under proper assumptions on ℎ, the problem has a local minimum around the origin with negative energy by Ekeland variational principle.Note that the term || −1  is neither 4-superlinear nor satisfies AR condition for  ∈ (1,3].In order to obtain the bounded PS sequence, we also use the indirect method in [7].Meanwhile, for  ∈  1 (R 3 ), we take a transform of   (⋅) = ( −2 ⋅) to construct the mountain pass geometry structure.Finally, the combination of Pohozaev identity with the method in [7] obtains the bounded PS sequence.Therefore, we obtain the second solution which has positive energy.
Our main result is as follows.
The paper is organized as follows.In Section 2, we give the existence of the negative energy solution  0 .The existence of positive energy solution V 0 and the proof of Theorem 1 are given in Section 3.

Existence of Negative Energy Solution
In this section, we give the existence of the negative energy solution.In order to obtain our first solution, we need the following preliminaries.Lemma 2. Let  ∈ (1, 5) and ℎ satisfy (h 1 ).Then, there exists ,  > 0 such that |   ⩾ , where   = { ∈  :‖  ‖< }.
Proof.Let {  } be a bounded PS sequence of , that is {  } and {(  )} are bounded,   (  ) → 0 in   , where   is the dual space of .We may assume that, up to a subsequence, It follows that By (8), we can obtain that Since {  } is also bounded in D 1,2 (R 3 ), then That is,   →  in .

Proof of Theorem 1
In this section, we will show the existence of the second solution.Note that  ∈ (1, 5), when  ∈ (1, 3], || −1  neither satisfies (AR) condition nor is 4-superlinear.So, in order to obtain the bounded PS sequence, following the argument in [6], we also use a direct method in [7].Firstly, we recall the following main result in [7].The "monotonicity trick" at the core of this theorem was invented by Struwe (see [9]).
Theorem 6 (see [7]).Let (, ‖ ⋅ ‖) be a Banach space and  ⊂ R + be an interval.Consider the family of  1 functionals on with  nonnegative and either () → ∞ or () → ∞ as ‖‖ → ∞.We assume that there are two points V 1 , V 2 in  such that where Then, for almost every  ∈  there is a sequence {  ()} ⊂  such that In our case,  = ,  = [1/2, 1], and define   :  → R by where  ∈ , Then {  } ∈ is a family of  1 functionals on .For any  ∈ , () ⩾ 0, and In the following, we verify that the functional   satisfies the conditions of Theorem 6.
(ii) Since   is nonincreasing on  ∈ , then by the definition of   and (i), for all  ∈ , we have In order to obtain the boundedness of {  }, we need the following Pohozaev identity.The proof is similar to the argument in [10].Lemma 8.Under the conditions of (h 1 ) and (h 2 ), if  ∈  is a weak solution of (1), the following Pohozaev identity holds: (3ℎ + (∇ℎ () , )) .