Multiple solutions of the Kirchhoff-type problem in R^N

This paper is to investigate the existence of multiple solutions for the following Kirchhoff-type problem

(1+b∫RN(|▿u|2+V(x)u2dx))[−▵u+V(x)u]=f(x,u), in RN$$\begin{array}{} \displaystyle \left( {1 + b{\smallint _{{R^N}}}\left( {{{\left| {\nabla u} \right|}^2} + V\left( x \right){u^2}dx} \right)} \right)\left[ { - \Delta u + V\left( x \right)u} \right] = f\left( {x,u} \right),{\rm{ \;\;in }}{R^N} \end{array}$$(1)

(where N = 1,2,3, constant b > 0. Set H := H¹(ℝ^N) = {u ∊ L²(ℝ^N) | ▽u ∊ L²(ℝ^N)} with the inner product (u,ν)_H = ∫_ℝ^N (▽u▽ν + uv)dx and the norm ||u||H=(u,u)H12.$\begin{array}{} \displaystyle {\left| {\left| u \right|} \right|_H} = \left( {u,u} \right)_H^{\frac{1}{2}}. \end{array}$.

In order to obtain that the variational structure of (1) and the corresponding variational functional I belongs to C¹(H,R), we need the some basic hypotheses such that f (x,u) ∊ C(R^N × R,R) and V ∊ C(R^N,R), f is superlinear near origin, subcritical growth; V has a positive infimum. Then, we can replace seeking the weak solutions of (1) with finding the critical points of the variational functional I (i.e. solving the Euler-Lagrange equations (1) of I).

Under these basic hypotheses, if set F(x,u):=∫0uf(x,s)ds$\begin{array}{} \displaystyle F\left( {x,u} \right): = {\rm{ }}\int_0^u {f\left( {x,s} \right)ds} \end{array}$, the variational functional of (1) is

I(u)=12||u||2+b4||u||4−∫RNF(x,u)dx$$\begin{array}{} \displaystyle I\left( u \right) = \frac{1}{2}{\left| {\left| u \right|} \right|^2} + \frac{b}{4}{\left| {\left| u \right|} \right|^4} - \smallint _{{R^N}}^{}{\rm{ }}F\left( {x,u} \right)dx \end{array}$$(2)

and it is of class C¹(see Lemma 4) on the real Sobolev space E := {u ∊ H | ∫R^N V (x)u²dx < + ∞}, which is equipped with the inner product and corresponding norm

(u,ν)=∫RN[▿u▿ν+V(x)uν]ⅆx,‖u‖=(u,ν)12.$$\begin{array}{} \displaystyle \begin{array}{*{20}{c}} {\left( {u,\nu} \right) = {\smallint _{{R^N}}}\left[ {\nabla u\nabla \nu + V\left( x \right)u\nu} \right]{\rm{d}}x,}&{u = {{\left( {u,\nu} \right)}^{\frac{1}{2}}}} \end{array}. \end{array}$$

Concretely speaking, we suppose that f and V satisfy the following assumptions:

(V₁) inf_x ∊ R^N V (x) ≥ V₀ ≤ 0, and for any M > 0, meas {x ∊ R^N : V (x) M} ≤ + ∞ where V₀ is a positive constant and meas denote the Lebesgue measure in R^N.

( f₁) f (x,u) = o(u) uniformly in x as |u| → 0.

( f₂) | f (x,u)| ≤ c(1 + |u|^p−1) for some c > 0 and p ∊ [2,2^*), where 2*={2NN−2,N≥3;+∞,N=1,2.$\begin{array}{} \displaystyle {2^*} = \left\{ \begin{array}{*{20}{l}} {\frac{{\begin{array}{*{20}{c}} {2N} \end{array}}}{{N - 2}},N \ge 3;}\\ { + \infty ,N = 1,2.} \end{array}\right. \end{array}$

( f₃) There exist μ > 4 and r > 0 such that

0<μF(x,u)≤uf(x,u)$$\begin{array}{} \displaystyle 0 < \mu F\left( {x,u} \right) \le uf\left( {x,u} \right) \end{array}$$(3)

for all u ∊ R\ {0} and x ∊ R^N and

Fr:=|r|−μ inf x∈RN,|u|=rF(x,u)>0.$$\begin{array}{} \displaystyle {F_r}: = {\left| r \right|^{ - \mu }}\mathop {{\rm{ \;inf }}}\limits_{x \in {R^N},\left| u \right| = r} F\left( {x,u} \right) > 0. \end{array}$$

(f3')$\begin{array}{} \displaystyle \left( {{{f'}_3}} \right) \end{array}$F(x,u)≤14uf(x,u)∀x∈RN,∀u∈R$\begin{array}{} \displaystyle F\left( {x,u} \right) \le \frac{1}{4}uf\left( {x,u} \right)\forall x \in {R^N},\forall u \in R \end{array}$

(f₄) F(x,u)u4→+∞$\begin{array}{} \displaystyle \frac{{F\left( {x,u} \right)}}{{{u_4}}} \to + \infty \end{array}$ uniformly in x as | u | → + ∞

(f₅)u↦f(x,u)|u|3$\begin{array}{} \displaystyle u \mapsto \frac{{f\left( {x,u} \right)}}{{{{\left| u \right|}^3}}} \end{array}$ is strictly increasing on (−∞,0) ∪ (0,+∞).

Set V (x) ≡ 0 and replace R^N by a bounded smooth domain Ω ⊆ R^N respectively, (1) reduces to the following Dirichlet problem of Kirchhoff type

{−(a+b∫Ω|▿u|2dx)▵u=f(x,u),inΩ;u=0,on∂Ω$$\begin{array}{} \displaystyle \left\{ \begin{array}{*{20}{c}} { - (a + b{\smallint _\Omega }|\nabla u{|^2}dx)\Delta u = f(x,u),} \hfill & {in{\rm{ \Omega }};} \hfill \\ {u = 0,} \hfill & {on{\rm{ }}\partial {\rm{\Omega }}} \hfill \\ \end{array}\right. \end{array}$$(4)

(4) is related to the stationary analogue of the equation

utt−(a+b)∫Ω|▿u|2dx)▵u=g(x,u)$$\begin{array}{} \displaystyle {u_{tt}} - \left( {a + b} \right){\smallint _{\rm{\Omega }}}{\left| {\nabla u} \right|^2}dx)\Delta u = g\left( {x,u} \right) \end{array}$$(5)

which was proposed by Kirchhoff [2] as an extension of the classical D’Alembert’s wave equation for free vibrations of elastic strings. The Kirchhoff’s model takes into account the changes in length of the string produced by transverse vibrations. It is pointed in [3] that (4) model several physical and biological systems, where u describes a process which depends on the average of itself (for example, population density). (5) received much attention only after Lions [4] introduced an abstract framework to it. Some interesting studies of (4) by variational methods can be found in [2, 3, 5–10].

Besides, (1) is variant of the following problem

−(a+b∫RN|▿u|2dx)▵u+V(x)u=f(x,u),x∈RN.$$\begin{array}{} \displaystyle \begin{array}{*{20}{c}} { - \left( {a + b{\smallint _{{R^N}}}{{\left| {\nabla u} \right|}^2}dx} \right)\Delta u + V\left( x \right)u = f\left( {x,u} \right),}&{x \in {R^N}.} \end{array} \end{array}$$(6)

Recently, problems like (1)(6) have been extensively studied by minimax theory in critical point theory, e.g. Mountain Pass Theorem [11] and Symmetric Mountain Pass Theorem [12], Linking Theorem and Fountain Theorem [13] and its variants [14] (see for example [15–18] for (6)).

Recently, some authors have studied (1) with perturbation methods( refer to [19, 20, 22, 23]). Particularly, Azzollini, d’Avenia and Pomponio [19] given a perturbation method and used it to obtain the multiple radial symmetric solutions of (1) with V (x) ≡ 0. Alves and Figueiredo [20] obtained two positive solutions of (1) by Nehari method, where V (x) is periodic, nonlocal term M(∫_R^N(|▽u|² + V(x)u²) is general and f (x,u) is subcritical or critical growth. By a general perturbed theorem in [21], Li, Li and Shi [22] got a positive solution of (1) provided that V (x) ≡ λ ≥ 0 and f (x,u) is independent of x combining cut-off functional with monotonicity trick. Ji [23] established the existence of infinitely many radially symmetric solutions of problem p(x)-(1) with radial potential via a direct variational method and the principle of symmetric criticality.

However, as far as we known, little has been done for the existence and multiplicity of nontrivial solutions of (1). Motivated by the above facts, this paper is to study the existence and multiplicity of nontrivial solutions of (1) by combining the direct method of the calculus of variation with minimax theory in critical point theory. Concretely, we shall find two distinct critical values of I: (1) a negative critical value of I natural constrained in a neighborhood of zero via Ekeland’s Variational Principle is obtained and (2) a positive critical value of I is obtained via the Mountain Pass Theorem.

Our results is as follows:

Remark 1. (V₁) is weaker than the coercivity of V , namely V (x) → + ∞ , as |x| ∞ +∞.

Remark 2. ( f₁)( f₅) not only implies (f3′)$\begin{array}{} \displaystyle \left( {f_3^\prime } \right) \end{array}$, but also that F(x,u) > 0 holds for u ≠ 0.

The remainder of this paper is organized as follows. Section 2 presents some preliminary results. Section 3 is devoted to the proof of results. Through out the paper, c_i and c are used in various places to denote positive constants.

Besides the multiple results of (1), we also take (1) as an example to give the main frame of finding the critical points of the variational functional I. The method of sequence convergence is a powerful tool to find the critical points of I by variational methods, the core idea (see [11, 13]) of which is essential as following:

Step 1. To obtain the variational functional I of (1) and prove that I ∊ C¹(H,R).

By Theorem 7.7 in [24], (1) has a variational functional I such as (2). In order to prove that I ∊ C¹(H,R), it is necessary to assume that f (x,u) ∊ C(R^N × R,R) and there is a positive constant c such that

|f(x,u)|≤c(|u|+|u|p−1),p∈[2,2*]$$\begin{array}{} \displaystyle \left| {f\left( {x,u} \right)} \right| \le c\left( {\left| u \right| + {{\left| u \right|}^{p - 1}}} \right),p \in \left[ {2,{2^*}} \right] \end{array}$$(7)

and V ∊ C(R^N,R) has a positive infimum V₀. ( f₁) is often called that f (x,u) is superlinear near origin, ( f₂) is often referred as that f (x,u) is subcritical growth, ( f₄) is also known as f (x,u) is 3-superlinear at ∞ and (3) is the famous (AR) condition. In fact, it is easy to see that ( f₁)( f₂) implies (7).

Step 2. To structure a good candidate for being a critical value of I.

(f₁)( f₂)(3) has been often used to show that I has a mountain pass geometry, namely that there are different points u₁,u₂ in H¹(ℝ^N) such that

c:=infγ∈Γ maxt∈[0,1]I(γ(t))>max{I(u1),I(u2)},$$\begin{array}{} \displaystyle c: = \mathop {{\rm{inf}}}\limits_{\gamma \in {\rm{\Gamma }}} \mathop {{\rm{ \;max}}}\limits_{t \in \left[ {0,1} \right]} I\left( {\gamma \left( t \right)} \right) > {\rm{max}}\left\{ {I\left( {{u_1}} \right),I\left( {{u_2}} \right)} \right\}, \end{array}$$

where Γ := {γ ∊ C([0,1],H)|γ(0) = u₁,γ(1) = u₂}, then the mountain pass level c is a good candidate for being a critical value of I. Furthermore, by the Deformation Lemma or the Ekeland’s Variational Principle, one sees that the mountain pass geometry directly implies the existence of a sequence {u_n} such that I(u_n) → c and I′(u_n) → 0. Such a sequence is called a Palais-Smale sequence at c ((PS)_c sequence for short).

In order to complete the first two steps, assumptions on f (x,u) should be ( f₁)( f₂) and ( f₄) rather than (7) and (3). In fact, ( f₁)( f₂)( f₄) are most essential because (3) implies ( f₄).

Step 3. To prove that (PS)_c condition holds, namely any (PS)_c sequence {u_n} possess a convergent subsequence.

In general, to establish (PS)_c condition, it suffices to insure the boundedness of (PS)_c sequence if I is weakly lower semicontinuous on H, I′(·) : H → H^* is compact and H is reflexive. In this process, (3) is also often used to obtain the boundedness of the (PS)_c sequence, and ( f₂) is also often combined with the following compactness of the embedding

H↪Lp(RN),p∈[2,2*)$$\begin{equation} H\hookrightarrow L^p(R^N),p\in [2, 2^*) \end{equation}$$(8)

to prove that I′(·) : H → H^* is compact and I is weakly lower semicontinuous.

Step 4. By the continuity of I and I′, I has a critical point u₀ at c.

To study (1) by using this method, there is general two difficulties.

One is that the dimension N must be restricted less than or equal to 3 if f satisfies ( f₂)( f₄). Under (3), several researchers studied (6). He and Zou [15] studied the existence, multiplicity and concentration behavior of positive solutions of (6) with N = 3 by using the variational methods. Wu [16] obtained the nontrivial solutions and a sequence of high energy solutions of (6) with N = 1,2,3 by using symmetric mountain pass theorem. Without (3), Liu and He [17] obtained infinitely many high energy solutions of (6) with N = 3 and sublinear f (x,u) = (p + 1)b(x)|u|^p−1u(0 < p < 1) combined a variant version of Fountain Theorem [14] and symmetric mountain pass Theorem to the Schrödinger operator eigenvalue theory.

Another is the lack of compactness of the embedding (8). Traditionally this difficult can be avoided by two class techniques. One is that, if V (x) satisfies (V₁), the compactness of embedding is regained by restricting the working space H to the subspace E. Another is that, if V (x) ≡ V₀ > 0 and f (x,u) depend only on |x| (namely which are radially symmetric), the compactness of embedding can be regained by restricting to the subspace H_r also. Jin and Wu [18] dealt with (6) with V (x) ≡ 1 in H_r, obtained infinitely many radial solutions. Moreover, if V (x) and f (x,u) = Q(x)|u|^p−1 are radially symmetric and satisfies some conditions in [25], the compactness of embedding of some weighted Sobolev spaces

Hr1(RN;V)↪Lp(RN;Q)$$H_r^1(R^N;V)\hookrightarrow L^p(R^N;Q)$$

is ensured also.

Let ||u||_p be the usual norm of the Lebesgue space L^p(R^N) . By Sobolev imbedding theorem (see [13, Theorem 1.8], the following embedding

H↪Lp(RN),p∈[2,2*)$$\begin{equation} H\hookrightarrow L^{p}(R^{N}), p \in [2,2^{*}) \end{equation}$$(9)

is continuous, that is, there are positive constants ν_p such that

||u||p≤νp||u||H,∀u∈H.$$\begin{array}{} \displaystyle {\left| {\left| u \right|} \right|_p} \le {\nu_p}{\left| {\left| u \right|} \right|_H},\forall u \in H. \end{array}$$

By (V₁), the embedding

E↪H$$\begin{equation} E\hookrightarrow H \end{equation}$$(10)

is continuous, that is, there is positive constant cν0=1min{1,Vo}$\begin{array}{} \displaystyle {c_{{\nu_0}}} = \frac{1}{{\sqrt {{\rm{min}}\left\{ {1,{V_o}} \right\}} }} \end{array}$ such that

||u||H≤cν0||u||,∀u∈E.$$\begin{array}{} \displaystyle {\left| {\left| u \right|} \right|_H} \le {c_{{\nu_0}}}\left| {\left| u \right|} \right|,\forall u \in E. \end{array}$$

By Lemma 3.4 in [27], (V₁) implies that the embedding

E↪Lp(RN),p∈[2,2*)$$\begin{equation} E\hookrightarrow L^{p}(R^{N}), p \in [2,2^{*}) \end{equation}$$(11)

is compact.

The following lemma shows that I has a mountain pass geometry on E: