Multiple positive solutions for a class of Kirchhoff type equations with indefinite nonlinearities

1 Introduction

We investigate the existence of multiple positive solutions to the Kirchhoff type equation with indefinite nonlinearities:

where N ≥ 3, a,b>0 , 1<q<2<p<min{4,2∗} , 2≤=2N/(N − 2), and functions k(x) and m(x) satisfy the following conditions.

(H₁) k ∈ C (ℝ^N) is a bounded function in ℝ^N;

(H₂) k is sign–changing in ℝ^N and Ω1={x∈RN:k(x)>0} is a bounded domain;

(H₃) m∈Lq∗(RN) and m⁺=max{m(x), 0} ≡ ≠ 0, where q∗=pp−q .

Problem (P) is a variant type of the Kirchhoff problem as follows:

which is related to the stationary analogue of the equation:

where Ω is a bounded domain in ℝ^N. Such problems are nonlocal owing to the appearance of ∫Ω|∇u|2dxΔu , which makes Eq. (1.1) and Eq. (1.2) are no longer pointwise identities. In [19], Kirchhoff firstly introduced Eq. (1.2) as an extension of the classical D'Alembert’s wave equation for free vibrations of elastic strings. The changes in length of the string produced by transverse vibrations are considered by the Kirchhoff’s model. Moreover, such nonlocal problem are also appeared in other fields, for instance, biological systems. For more details about the backgrounds of problems (1.1) and (1.2), one can be referred to [2,4,12,20].

In [22], Lions firstly proposed an abstract framework of Eq. (1.1), from then on, lots of researchers began to study Eq. (1.1) in general dimension, see [11,16,25,26,28,39] and the references therein. More precisely, Zhang and Perera [39] obtained the existence of a positive solution, a negative solution and a sign–changing solutions for Eq. (1.1) with N ≥ 1 by using invariant sets of descent flow. Pei and Ma [26] got three positive solutions for Eq. (1.1) with N=1, 2, 3 via the minimax method and the Morse theory. By using the theory developed in [27], Ricceri [28] obtained three positive solutions for Eq. (1.1) with N ≥ 4.

Recently, many papers [6, 8,29,30,38,32,15,31] study the Kirchhoff equation on the whole space:

When the potential function V(x) satisfies the following assumptions:

(V₁) V ∈ C (ℝ^N) with V(x) ≥ 0 in ℝ^N and there exists c0>0 such that the set {V<c0}:={x∈RN| V(x)<c0} has finite positive Lebesgue measure, where ∣·∣ is the Lebesgue measure;

(V₂) Ω=int{x∈RN ∣ V(x)=0} is nonempty and has smooth boundary with Ω‾=V−1(0) . Sun and Wu [29] obtained the existence, nonexistence and concentration of nontrivial solutions for Eq. (1.4) with N=3 and V being replaced by λV, λ>0 . Later, when N ≥ 4, V is replaced by λV, λ>0 , and f satisfies the superlinear condition, Sun et al. [30] proved that Eq. (1.4) possessed two positive solutions. When N=3, V satisfies (V₁)−(V₂) and f satisfies the classical Ambrosetti–Rabinowitz type condition, Xie and Ma [38] proved that Eq. (1.4) has at least a positive solution. Moreover, the concentration behavior is also studied by the authors. In [32], Sun and Zhang obtained the uniqueness of the positive ground state solution for Eq. (1.4) with N=3, f(x,u)=d|u|p−1u, 3<p<5 , and V(x) ≡ c, where c and d are positive constants. Besides, by using the uniqueness result and the concentration–compactness lemma [23], the authors also got the existence and concentration theorems for the following Kirchhoff problem:

In [15], by using the Schwartz symmetric arrangement, Guo proved that Eq. (1.4) possesses a positive ground state solution when V is continuous and f does not satisfies the classical Ambrosetti–Rabinowitz type condition.

For the elliptic problems with concave–convex nonlinearities, there are also several results. For example, one can be referred to [1,37,35,24,36] and the references therein. Indeed, in [1], Ambrosetti et al. firstly introduced the elliptic problem involving the concave–convex nonlinearities:

where 1<q<2<p≤2∗ (2≤=∞ if N=1, 2; 2≤=2N/(N − 2) if N ≥ 3), λ>0 and Ω⊂ℝ^N(N ≥ 1) is bounded. With the aid of variational methods, the authors established the existence, nonexistence and multiplicity of solutions for Eq. (1.6). Later, Wu [37] obtained the existence of multiple positive solutions for the following problem:

where 1<q<2<p≤2∗ (2≤=∞ if N=1, 2; 2≤=2N/(N − 2) if N ≥ 3), f is sign–changing and g is positive in ℝ^N.

To the best of our knowledge, for the Kirchhoff problem with concave–convex terms, there are few results [5, 8,9,10, 21]. In [9], Chen et al. considered a class of Kirchhoff problem on the bounded domain Ω⊂ℝ^N(N ≥ 1):

where 1<q<2<p≤2∗ (2≤=∞ if N=1, 2; 2≤=2N/(N − 2) if N ≥ 3). By giving different scopes on a and λ, the authors proved that Eq. (1.8) possesses multiple positive solutions. By using the Nehari manifold technique, Liao et al. [21] obtained two nonnegative solutions for Eq. (1.8). When f(x)=g(x) ≡ 1 in Eq. (1.8), by constraining the energy functional of the problem on a subset of the nodal Nehari set, Chen and Ou [10] proved that there exists a constant λ∗>0 such that for any λ<λ∗ , Eq. (1.8) has a nodal solution u_λ with positive energy. In [8], we obtained three positive solutions for the Kirchhoff type problem with steep potential well.

Motivated by the above facts, more precisely by [9], it is quite natural for us to ask that whether Eq.(P) can have multiple positive solutions when 1<q<2<p≤min{4,2∗} and the domain is unbounded? As far as we know, such problem has never been discussed in the available literature. In our paper, we will give a definite answer to the above question. Under some suitable conditions on functions k and m, we will obtain three positive solutions for Eq.(P). It is worthy pointing out, in [8], we considered Eq.(P) with steep potential well and positive nonlinearities, which we overcome the compactness of the Sobolev embedding by giving some inequalities. While, in the present paper, since the potential function is a constant and the nonlinearities are sigh–changing, the standard method of recovering the compactness does not work, motivated by [17], a novel method will be used to get the Palais–Smale(PS for short) condition.

Before introducing our main conclusions, we first recall a well known result (c.f. [34]). Suppose that wΩ1 is the positive ground state solution to the nonlinear elliptic equation:

where Ω₁ is defined by the condition (H₂).

Obviously,

and

where JΩ1 is the energy functional of problem (EΩ1) , defined by

and the corresponding Nehari manifold is given by:

In order to simplify the calculation, in the present paper, we hypothesize that a=1 in problem (P). Denote kmax:=supx∈Ω1‾k(x) and A(p)=24−p1/(p−2) . Let S_r, Sr,Ω1 and S be the best Sobolev constants for the embedding of H¹(ℝ^N) in L^r(ℝ^N), H01(Ω1) in L^r(Ω₁) and D^1,2(ℝ^N) in L2∗(RN) with 2 ≤ r ≤ 2≤, respectively. For any 2 ≤ r ≤ +∞, we shall also denote by ∣·∣_r the L^r–norm. If we take a subsequence of a sequence {u_n}, we may denote it again by {u_n},

We now summarize our main results below.

2 Preliminaries

For any u ∈ H¹(ℝ^N), let

Then I_b is a well defined C¹ functional with the following derivative:

Define the Nehari manifold

Thus u∈Mb if and only if

Note that Mb is closely related to the behavior of the fibering map [13,3]: hb,u:t→Ib(tu),t>0 . From (2.1), it is easy to know that

Thus

and

A straightforward calculation shows that

Obviously, tu∈Mb holds if and only if h′_{b, u}(t)=0. Hence, points in Mb correspond to the stationary points of the maps h_{b, u}, then it is natural to divide Mb into three parts corresponding to the local minima, local maxima and points of inflection. Following [33], we define

and

Similar to the arguments of Brown and Zhang [3, Theorem 2.3], we may get the following conclusion.

3 Existence of two negative–energy positive solutions