Journal of Applied Mathematics and Computing

, Volume 39, Issue 1, pp 473–487

Multiplicity of high energy solutions for superlinear Kirchhoff equations

Original Research

DOI: 10.1007/s12190-012-0536-1

Cite this article as:
Liu, W. & He, X. J. Appl. Math. Comput. (2012) 39: 473. doi:10.1007/s12190-012-0536-1

Abstract

In this paper, we study the existence of infinitely many high energy solutions for the nonlinear Kirchhoff equations
$$\left\{\everymath{\displaystyle}\begin{array}{l@{\quad}l}- \biggl(a+b\int_{R^3} |\nabla u|^2 dx\biggr)\Delta u + V(x)u=f(x,u),&x\in \mathbb {R}^3,\\[9pt]u\in H^1 (\mathbb {R}^3),\end{array}\right.$$
where a,b>0 are constants, V:ℝ3→ℝ is continuous and has a positive infimum. f is a subcritical nonlinearity which needs not to satisfy the usual Ambrosetti-Rabinowitz-type growth conditions.

Keywords

Nonlinear Kirchhoff equations High energy solutions Variational methods 

Mathematics Subject Classification (2000)

35J60 35J25 

1 Introduction and preliminaries

We consider the following nonlinear Kirchhoff equations
$$ \left\{\everymath{\displaystyle}\begin{array}{l@{\quad}l}-\biggl(a+b\int_{R^3} |\nabla u|^2 dx\biggr)\Delta u + V(x)u=f(x,u),& x\in \mathbb {R}^3,\\[9pt]u\in H^1 (\mathbb {R}^3),\end{array}\right.$$
(1.1)
where a,b>0 are constants, VC(ℝ3,ℝ) and f(x,u)∈C(ℝ3×ℝ,ℝ) satisfy some further conditions.
In (1.1), if we set V(x)≡0 and replace ℝ3 by a bounded domain Ω⊂ℝN, then (1.1) reduces to the following Dirichlet problem of Kirchhoff type:
$$ \left\{\everymath{\displaystyle}\begin{array}{l@{\quad}l}-\biggl(a+b\int_{\Omega}|\nabla u|^2dx\biggr)\Delta u=f(x,u)&\mbox{in}\ \Omega,\\[9pt]u=0&\mbox{on}\ \partial\Omega.\end{array}\right.$$
(1.2)
Problem (1.2) is related to the stationary analogue of the equation
$$ u_{tt}-\biggl(a+b\int_{\Omega}|\nabla u|^2dx\biggr)\Delta u=f(x,u)$$
(1.3)
proposed by Kirchhoff in [1] as an existence of the classical D’Alembert’s wave equations 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 classical investigations of Kirchhoff equations can be seen in Bernstein [2] and Pohoẑaev [3]. Equation (1.3) received much attention only after Lions [4] introduced an abstract framework to the problem. Some interesting results can be found, for example, in [5, 6, 7, 8]. In [5], Arosio and Panizzi studied the Cauchy-Dirichlet type problem related to (1.3) in the Hadamard sense as a special case of an abstract second-order Cauchy problem in a Hilbert space. In [6], Cavalcanti, Cavalcanti and Soriano considered the question of the existence and uniqueness of regular global solutions for the Kirchhoff-Carrier equation subject to nonlinear boundary dissipation without restriction on the initial data and obtained uniform decay rates by assuming a nonlinear feedback acting on the boundary. In [8], D’Ancona and Spagnolo proved the existence of a global classical periodic solution for the degenerate Kirchhoff equation with real analytic data. In particular, in the paper [8], Kirchhoff’s equation is an example of a quasi-linear hyperbolic Cauchy problem that describes the transverse oscillations of a stretched string. We also note that several existence results have been obtained for (1.2) (on bounded domain) in recent years. For example, Alves, Corrêa and Ma [9] studied problem (1.2) and obtained positive solutions via the Mountain Pass Theorem. Ma and Rivera [10] obtained positive solutions of such problems by using variational methods. Perera and Zhang [11] proved the existence of one nontrivial solution to (1.2) via Yang index and critical group. Zhang and Perera [12], Mao and Zhang [13] obtained three solutions (a positive solution, a negative solution and a sign-changing solution) by the invariant sets of descent flow. He and Zou [14] showed the existence of infinitely many solutions by using the local minimum methods and the fountain theorems. Cheng and Wu [15] studied the existence of positive solutions for problem (1.2) when the nonlinearity f is asymptotically t3-growth at infinity. We refer to [16, 17, 18, 19] for more existence results of the Kirchhoff-type equations.
We remak that when a=1,b=0, problem (1.1) can be rewritten as the following well-known Schrödinger equation:
$$ -\Delta u+V(x)u=f(x,u),\quad x\in \mathbb {R}^N.$$
(1.4)
For (1.4), there is a large quantity of study on the existence and multiplicity of solutions in the literature, we refer to [20, 21, 22, 23, 24, 25] and the references therein.
The main purpose of this paper is to study the existence of infinitely many high energy solutions for the nonlinear Kirchhoff equation (1.1). Before stating our main results, we first make some assumptions on the functions V and f. For the potential V we make the following assumption.
  1. (V)
    \(V(x)\in L^{\infty}_{loc}(\mathbb {R}^{3}),V_{0}:=\inf_{\mathbb {R}^{3}}V(x)>a_{1}>0\). For any M,r>0,
    $$m(\{x\in B_r(y):~V(x)\leq M\})\rightarrow 0\quad\mbox{as}\ |y|\rightarrow\infty,$$
    where Br(y) denotes the ball centered at y with radius r.
     

Remark 1.1

The condition (V) was first introduced by Bartsch and Wang [21] to ensure the compactness of embeddings of the work spaces. The limit in condition (V) can be replaced by one of the following simpler conditions:
  1. (V1)

    meas(xR3:V(x)≤M)<∞ for any M>0 (cf. [21]),

     
  2. (V2)

    V(x)→∞ as |x|→∞ (cf. [24]).

     
The standard elliptic theory (cf. [24]) implies that the eigenvalue problem
$$ -\Delta u+V(x) u=\lambda u,\quad x \in R^3,$$
(1.5)
possesses a sequence of positive eigenvalues: 0<λ1<λ2<⋯<λk<⋯→∞ with finite multiplicity for each λk. The principal eigenvalue λ1 is simple with positive eigenfunction φ1, and eigenfunction φk corresponding to λk(k≥2) is sign-changing.
For the nonlinearity f, we suppose it satisfies the following conditions in this paper.
(f1)

\(\lim_{|s| \rightarrow0} \frac{f(x,s)}{s} = 0\) uniformly for x∈ℝ3.

(f2)
fC(ℝ3×ℝ,ℝ),f(x,s)s≥0, for s≥0; and there exists a2>0,q∈(2,2) such that
$$|f(x,s)| \leq a_2 (1+|s|^{q-1}),\quad\mbox{for all}\ x \in \mathbb {R}^3, s\in \mathbb {R},$$
where 2=6 is the critical exponent for the Sobolev embedding in dimension 3.
(f3)
There exists μ>4 and R>0 such that
$$\mu F(x,s)\leq sf(x,s),\quad\mbox{for all}\ (x,s)\in \mathbb {R}^3\times \mathbb {R}\ \mbox{and}\ |s|\geq R.$$
(f4)

There exists 4<α<2 such that \(\lim\inf _{|s|\rightarrow\infty}\frac{F(x,s)}{|s|^{\alpha}}>0\), uniformly in x∈ℝ3.

(f5)

f(x,−s)=−f(x,s) for any x∈ℝ3,s∈ℝ.

We fix the following notations. For any 1≤s<∞, Ls(ℝ3) denotes the usual Lebesgue space with the norm \(\|u\|_{L^{s}} := (\int_{R^{3}} |u|^{s}dx)^{\frac{1}{s}}\). H1(ℝ3) is the usual Sobolev space with the norm
$$\|u\|:=\left(\int_{R^3} (|\nabla u|^2 +u^2)dx\right)^ \frac{1}{2}.$$
In our problem, the work space E is defined by
$$E:=\left\{u \in H^1 (\mathbb {R}^3)\bigg|~\int_{R^3}(|\nabla u|^2 + V(x)u^2)dx<\infty\right\}.$$
Thus, E is a Hilbert space with the inner product \((u,v)_{E} :=\int_{R^{3}} (\nabla u \nabla v + V(x)uv)dx\), and its norm is \(\|u\|_{E} = (u,u)_{E} ^{\frac{1}{2}}\). Since V(x) is bounded form below, the embedding ELs(ℝ3) is continuous for any s∈[2,2].
It is well-known that weak solutions of (1.1) are precisely the critical point of the functional

The main results are the following.

Theorem 1.1

Assume that conditions (V), (f1)–(f5) hold. Then problem (1.1) has infinitely many solutions {un} satisfyingunE→∞ and Φ(un)→∞ asn→∞.

Theorem 1.2

In addition to conditions (V), (f1)–(f2) and (f5), suppose thatfsatisfies the following conditions
(f3)′

\(\lim_{|s| \rightarrow\infty} \frac{f(x,s)s}{s^{4}} = \infty\)uniformly forx∈ℝ3.

(f4)′

For a.e. x∈ℝ3,∀(s,t)∈ℝ+×ℝ+,st, there holds\(\mathcal{G}(x,s) \leq\mathcal{G}(x,t)\), where\(\mathcal{G}:\mathbb {R}^{3} \times \mathbb {R}^{+} \rightarrow \mathbb {R}\)is defined by\(\mathcal{G}(x,s):= \frac{1}{4} f(x,s)s -F(x,s)\).

Then problem (1.1) has a sequence of solutions {un} inEwithunE→∞ and Φ(un)→∞ asn→∞.

Clearly, dealing with Φ, one has to face various difficulties: we mention that the competing effect of the non-local term with the nonlinear term gives rise to very different situations and that the lack of compactness of the embedding of H1(R3) in the Lebesgue spaces Lp(R3),p∈(2,6), prevents from using the variational techniques in a standard way. This last difficulty can be avoided, if we restrict Φ to the subspace of H1(R3) consisting of radially symmetric functions, or, when one is searching for semi-classical states, by using perturbation methods or a reduction to a finite dimension by the projections method. As far as we know, except a perturbation result in [26], there are no existence results when the potential and the coefficient of the nonlinearity are not symmetric and a not singularly perturbed problem is considered.

In order to obtain infinitely many high energy solutions of problem (1.1), we shall apply Theorem 9.12 [27] and a variant version of fountain theorem [24], which will be stated later. The proofs of Theorems 1.1 and 1.2 are carried out in Sects. 2 and 3, respectively.

Hereafter, the letters C,Ci,i=1,2,…, will be repeatedly used to denote various constants whose exact values are irrelevant.

2 Proof of Theorem 1.1

Let Xi denote the eigenspace of λi; then dimXi<∞. Let E1=X1X2⊕⋯⊕Xk and
$$E_2=\overline{\bigoplus_{i=k+1}^\infty X_i}.$$
In such a way, E has a direct sum decomposition E=E1E2 with dimE1<∞. Under the conditions of Theorem 1.1, it is easy to see that Φ∈C1(E,ℝ).

Definition 2.1

Any sequence {un}⊂E satisfying
$$\sup_n|\Phi(u_n)|<\infty,\qquad\Phi'(u_n)\rightarrow 0,$$
is called a Palais-Smale sequence ((PS) sequence, for short). If any (PS) sequence of Φ possesses a convergent subsequence, we say that Φ satisfies the (PS) condition.

In order to obtain infinitely many solutions of (1.1), we shall use the following critical point theorem [27] which was introduced by Rabinowitz.

Lemma 2.1

([27], Theorem 9.12)

LetEbe an infinite dimensional real Banach space andIC1(E,ℝ) be even, satisfy the (PS) condition andI(0)=0. IfE=E1E2withE1being finite dimensional, andIsatisfies
  1. (i)

    there exist constantsρ,α>0 such that\(I|_{\partial B_{\rho}\cap E_{2}}\geq\alpha\), where∂Bρ={uE:∥u∥=ρ}, and

     
  2. (ii)

    for each finite dimensional subspace\(\widehat{E}\subset E\), there exists an\(r=r_{\widehat{E}}>0\)such thatI≤0 on\(\widehat{E}\backslash B_{r}\).

     
Then, Ipossesses an unbounded sequence of critical values.

Under condition (V), we have the following compactness result.

Lemma 2.2

[21]

Under the assumption (V), the embeddingELs(ℝ3),2≤s<2is compact.

If we replace (V) by (V1) or (V2), then the conclusion of Lemma 2.2 is still true.

Lemma 2.3

If conditions (f1),(f2),(f3) and (f4) hold. Then any (PS) sequence of Φ is bounded inE.

Proof

Let {un}⊂E be a (PS)c sequence, that is Φ(un)=c+on(1) and Φ′(un)→0. Suppose by contradiction that, {un} is unbounded in E. Set \(v_{n}=\frac{u_{n}}{\|u_{n}\|_{E}}\), then ∥vnE=1. Since In view of α>4, we see that
$$\lim_{n\rightarrow\infty}\int_{\mathbb {R}^3}\frac{f(x,u_n)u_n}{\|u_n\|_E^{\alpha}}dx=0.$$
On the other hand, since ∥vnE=1, we can find some vE such that vnv in E;vnv in Ls(ℝ3) for s∈[2,2) and vnv a.e. x∈ℝ3. Denote \(\Omega=\{x\in \mathbb {R}^{3}:~v(x)\not=0\}\). If meas(Ω)>0, then |vn(x)|>0 a.e. x∈Ω. By (f1),(f2),(f3) and (f4), we have
$$f(x,u)u\geq C_1|u|^\alpha-C_2u^2$$
for all (x,u)∈ℝ3×ℝ. So
$$\int_{\mathbb {R}^3}\frac{f(x,u_n)u_n}{\|u_n\|_E^\alpha}dx\geq C_1\|v_n\|^\alpha_\alpha-C_2\frac{\|v_n\|^2_2}{\|u_n\|_E^{\alpha -2}}. $$
(2.1)
From (2.1), we have
$$0=\lim_{n\rightarrow\infty}\int_{\mathbb {R}^3}\frac{f(x,u_n)u_n}{\|u_n\|_E^\alpha}dx\geq C_1\|v\|^\alpha_\alpha=C_1\int_{\Omega}|v|^\alpha dx>0,$$
which leads to a contradiction. Therefore, meas(Ω)=0, and so, v(x)=0 a.e. x∈ℝ3. By (f1),(f2) and (f3), there is a constant C3>0 such that
$$uf(x,u)-\mu F(x,u)\geq-C_3u^2$$
for all (x,u)∈ℝ×ℝ. Consequently, Taking limit in (2.2), we get \(0\geq(\frac{1}{2}-\frac{1}{\mu})\min\{a,1\}>0\), a contradiction. Therefore, {un} is bounded in E. □

Lemma 2.4

Under the conditions of Theorem 1.1, the functional Φ satisfies the Palais-Smale condition.

Proof

Suppose that {un} is a (PS) sequence of Φ, then by Lemma 2.3 we see that {un} is bounded in E. Therefore, we can extract a subsequence, still denoted by {un} such that unu in E and unu in Ls(ℝ3),s∈[2,2). We note that From (2.3), we have Using the boundedness of {un} and unu in E, one has
$$b\left(\int_{\mathbb {R}^3}(|\nabla u|^2-|\nabla u_n|^2)dx\right)\int_{\mathbb {R}^3}\nabla u\nabla(u_n-u)dx\rightarrow 0,\quad\mbox{as}\ n\rightarrow\infty. $$
(2.5)
By (f1),(f2), for any given ε>0, there exists Cε>0 such that
$$|f(x,u)|\leq\varepsilon|u|+C_\varepsilon|u|^{q-1},\quad\forall(x,u)\in \mathbb {R}^3\times \mathbb {R}.$$
Therefore, where C are positive constants independent of ε,n. Since ∥unu2→0,∥unuq→0 as n→∞, we see that
$$\int_{\mathbb {R}^3}(f(x,u_n)-f(x,u))(u_n-u)dx\rightarrow 0,\quad\mbox{as}\ n\rightarrow\infty. $$
(2.7)
Since Φ′(un)→0, we deduce from (2.4), (2.5) and (2.7) that ∥unuE→0, i.e., the (PS) condition holds. □

Proof of Theorem 1.1

The rest proof of Theorem 1.1 can be completed by dividing it into the following steps:
Step 1.
There exist constants ρ>0 and α>0 such that \(\Phi|_{\partial B_{\rho}\cap E_{2}}\geq\alpha\). We fix a number k∈ℕ and choose λ∈[λk,λk+1). Then by (f1), there exists δ>0 such that
$$|f(x,s)|\leq\lambda\min\{a,1\}|s|\quad\mbox{for all}\ x\in \mathbb {R}^3, |s|\leq\delta.$$
Therefore, by (f2), we have
$$F(x,s)\leq\frac{\min\{a,1\}}{2}\lambda s^2+C_4|s|^q,\quad\mbox{for all}\ (x,s)\in \mathbb {R}^3\times \mathbb {R}.$$
For uE2, we have Therefore, there exists some small ρ>0 such that
$$\Phi(u)\geq\frac{\min\{a,1\}}{4}\left(1-\frac{\lambda}{\lambda _{k+1}}\right)\rho^2:=\alpha>0$$
for all uE2 with ∥uE=ρ.
Step 2.
For any finite dimensional space \(\widehat{E}\subset E\), we have that Φ(u)≤0 for all \(u\in\widehat{E}\backslash B_{r}\) with some \(r=r_{\widehat{E}}>0\). In fact, by (f1),(f2) and (f3), there exist positive constants C6,C7 such that
$$F(x,u)\geq C_6|u|^{\mu}-C_7|u|^2$$
for all (x,u)∈ℝ3×ℝ. Thus, for each \(u\in\widehat{E}\backslash B_{r}\), we have Since \(\widehat{E}\subset E\) is a finite dimensional space, all the norms on \(\widehat{E}\) are equivalent, we can choose \(r=r_{\widehat{E}}>0\) such that
$$I(u)\leq0,\quad\mbox{for all}\ u\in\widehat{E}\backslash B_r.$$

Now, by Lemma 2.4, Φ satisfies (PS) condition. Moreover, Φ is even due to (f5). Thus, the conclusion follows from Lemma 2.1.

3 Proof of Theorem 1.2

In order to prove Theorem 1.2, we shall use a variant version of fountain theorem due Zou [24], which can be stated as the following.

Let X be a Banach space with the norm ∥⋅∥ and let Xi,i∈ℕ be a sequence of subspace of X with dimXj<∞ for each j∈ℕ. Further, \(X=\overline{\bigoplus_{j\in \mathbb {N}}X_{j}}\), the closure of the direct sum of all Xj. Set \(Y_{k} =\bigoplus_{j=0}^{k} X_{j}, Z_{k}=\overline{\bigoplus_{i=k}^{\infty}X_{i}}\) and Bk={uYk:∥u∥≤ρk},Sk={uZk:∥u∥≤γk}, for ρk>γk>0. Consider a family of \(\mathcal {C}^{1}\)-functional Φλ(u):X→ℝ of the form:
$$\Phi_\lambda(u) : = A(u)-\lambda B(u), \quad\lambda\in [1,2].$$
We make the following assumptions.
(H1)

Φλ(u) maps bounded sets into bounded sets uniformly for all λ∈[1,2]. Furthermore Φλ(−u)=Φλ(u) for all (λ,u)∈[1,2]×X.

(H2)

B(u)≥0 for all uX;A(u)→∞ or B(u)→∞ as ∥u∥→∞, or

(H3)

B(u)≤0 for all uX;B(u)→−∞ as ∥u∥→∞.

Lemma 3.1

([24], Theorem 3.1)

Assume (H1) and (H2) (or (H3)) hold. Ifbk(λ)>ak(λ) for allλ∈[1,2], thenck(λ)≥bk(λ) for allλ∈[1,2]. Moreover, for a.e. λ∈[1,2], there exists a sequence\(\{u_{n} ^{k} (\lambda)\}_{n} ^{\infty}\)such that
$$\sup_n \|u_n^k (\lambda)\| <{\infty},\qquad\Phi_{\lambda}' (u_n^k(\lambda)) \rightarrow0\quad\mathit{and}\quad\Phi_{\lambda} (u_n^k(\lambda)) \rightarrow c_k ({\lambda})\quad\mathit{as}\ n \rightarrow {\infty}.$$
Let E be the Banach space given in the previous section and Xj:=span{ej}, where ej is the eigenfunction corresponding to the eigenvalue λj. Denote \(Y_{k} = \bigoplus_{j=0} ^{k} X_{j}\), \(Z_{k} = \overline{\bigoplus _{j=k} ^{\infty}X_{j}}\). Consider the family of functional Φλ:E→ℝ defined by Then under conditions of Theorem 1.2, B(u)≥0 for all uE; A(u)→∞ as ∥uE→∞, and Φλ(−u)=Φλ(u) for all (λ,u)∈[1,2]×E due to (f5). And it is easy to see that Φλ maps bounded sets to bounded sets uniformly for λ∈[1,2].

Now, we show that Φλ satisfies the geometric properties of Lemma 3.1.

Lemma 3.2

Suppose that (f1),(f2) and (f3)′ are satisfied. Then there existsρk>γk>0 such that
  1. (i)

    \(a_{k}(\lambda):= \max_{u \in Y_{k}, \|u\|_{E}= \rho_{k}} \Phi _{\lambda} (u) \leq 0\),

     
  2. (ii)

    \(b_{k}(\lambda):= \inf_{u \in Z_{k}, \|u\|_{E}= \gamma_{k}}\Phi _{\lambda}(u)>0\).

     

Proof

(i) By (f3)′, for any L>0, there exists δ=δ(L)>0, such that for all x∈ℝ3, |u|≥δ, we have f(x,u)uLu4 and
$$F(x,u) \geq\frac {L}{4} u^4. $$
(3.2)
From (f1), we see that there exists β>0 small, such that
$$|f(x,u)u| \leq u^2 $$
(3.3)
for all x∈ℝ3, 0<|u|≤β. By (f2), there exists some L1=L1(δ,L) such that for all x∈ℝ3,β≤|u|≤δ,
$$\left|\frac{f(x,u)u}{u^2}\right| \leq\frac {a_2(1+|u|^{q-1})|u|}{u^2} \leq L_1. $$
(3.4)
From (3.4) and (3.3), for all x∈ℝ3, 0<|u|≤δ we have f(x,u)u≥−(L1+1)|u|2, and hence
$$F(x,u) \geq-\frac{1}{2}(L_1+1)u^2. $$
(3.5)
Put \(M:=\frac{1}{2}(L_{1}+1)+\frac{1}{4}L \delta^{4}\). Combining (3.2) and (3.5), we have for all x∈ℝ3,u∈ℝ,
$$F(x,u) \geq\frac{1}{4}Lu^4 -Mu^2.$$
Hence, for any uYk, we obtain where in the last inequality we have used the equivalence of all norms on the finite dimensional subspace Yk.
Now we take L large enough such that
$$\frac{b}{4}-\frac{\lambda}{4}LC <0.$$
Thus, we can take ∥uE=ρk>0 large enough such that
$$a_k(\lambda):= \max_{u \in Y_k, \|u\|_{E}= \rho_k}\Phi_{\lambda} (u) \leq0.$$
(ii) It follows from (f1) and (f2) that, for any ξ>0, there exists Cξ>0 such that
$$F(x,u) \leq\xi u^2+C_{\xi} |u|^q,\quad\mbox{for any}\ x \in \mathbb {R}^3,~u \in \mathbb {R}. $$
(3.6)
Let
$$\beta_k := \sup_{u \in Z_k,~\|u\|_E=1} {\|u\|_{p}}$$
for any 2<p<2. Then βk→0, as k→∞ (cf. [22]). For simplicity, let a:=min(a,1), then for any uZk and ξ>0 small enough, we have Denote \(\gamma_{k} = (\lambda C_{\xi}\frac {q}{a_{\star}}\beta_{k}^{q})^{\frac{1}{2-q}}\), then Since βk→0 as k→∞,q>2 and ξ can be small arbitrarily, we have
$$b_k(\lambda) \geq\widetilde{ {b_k}} \rightarrow\infty\quad\mbox{as}\ k \rightarrow\infty. $$
(3.7)
 □
At this stage, we see that the conditions of Lemma 3.1 are satisfied. Hence, for all most every λ∈[1,2], there exists a sequence \(\{u_{n}^{k}(\lambda)\}_{n=1}^{\infty}\) such that where \(c_{k}(\lambda)=\inf_{\gamma\in\Gamma_{k}} \max_{u \in B_{k}}\Phi_{\lambda} (\gamma(u))\). Note that \(c_{k}(\lambda) \leq\sup_{u\in B_{k}} \Phi_{1} (u) := \widetilde{{c_{k}}}\) and the embedding ELs(ℝ3), 2≤s<2 is compact. By a similar argument as we have done in the proof of Theorem 1.1, \(\{u_{n}^{k}(\lambda )\}_{n=1}^{\infty}\) has a convergent subsequence. Suppose \(u_{n}^{k}(\lambda) \rightarrow u^{k}(\lambda)\) as n→∞. Then we have
$$\Phi_{\lambda}' (u^k(\lambda))= 0,\qquad\Phi_{\lambda} (u^k(\lambda)) \in[\widetilde{{b_k}},\widetilde{{c_k}}],\quad\mbox{for almost every}\ \lambda \in[1,2].$$
Now we take a sequence {λn}⊂[1,2] with λn→1, and set un:=uk(λn) for simplicity. Then \(\Phi_{\lambda_{n}}'(u_{n})=0, \Phi_{\lambda_{n}} (u_{n})\in[\widetilde{{b_{k}}},\widetilde{{c_{k}}}]\).

Lemma 3.3

Under the assumptions of Theorem 1.2, the sequence {un} is bounded in E.

Proof

Suppose by contradiction that, ∥unE→∞ as n→∞. Define \(\omega_{n}:=\frac{u_{n}}{\|u_{n}\|_{E}}\). Then ∥ωnE=1,∀n∈ℕ. Passing to a subsequence, and using the compact embedding ELs(ℝ3),2≤s<2, we obtain
$$ \left\{\begin{array}{l@{\quad}l}\omega_n\rightharpoonup v&\mbox{weakly in } E,\\[4pt]\omega_n\rightarrow v&\mbox{strongly in } L^s(\mathbb {R}^3), 2\leq s <2^*,\\[4pt]\omega_n\rightarrow v&\mbox{a.e. }x\in \mathbb {R}^3.\end{array}\right.$$
(3.8)
Next we distinguish the two possible cases: (i) v≠0 and (ii) v=0.
In case (i), by \(\Phi_{\lambda_{n}}'(u_{n})=0\), we have Dividing both sides of the above equality by \(\lambda_{n}\|u_{n}\|_{E}^{4}\), we obtain
$$\int_{R^3} \frac{ f(x,u_n)u_n}{\|u_n\|_E^4}dx \leq\frac {\max\{a,1\}}{\lambda_n \|u_n\|_E^2}+\frac{b}{\lambda_n} \leq C <\infty. $$
(3.9)
On the other hand, on the set Σ:={x∈ℝ3v(x)≠0},
$$\frac{f(x,u_n)u_n}{\|u_n\|_E^4} = \frac{ f(x,u_n)u_n}{u_n^4}\cdot\omega_n^4 (x) \rightarrow\infty\quad\mbox{as }n \rightarrow \infty.$$
Since |Σ|>0, applying Fatou’s lemma we conclude
$$\lim_{n \rightarrow\infty} \int_{R^3} \frac{f(x,u_n)u_n}{\|u_n\|_E^4} dx\geq \int_{\Sigma}\frac{ f(x,u_n)u_n}{u_n^4}\cdot\omega_n^4 (x) dx = \infty,$$
which contradicts to (3.9).
In case (ii), we can define as in [28],
$$\Phi_{\lambda_n} (t_nu_n):= \max_{t \in[0,1]} \Phi_{\lambda_n}(tu_n).$$
For any given M>0, define \(\widetilde{{\omega_{n}}}=\sqrt{4M}~\frac{u_{n}}{\|u_{n}\|_{E}} =\sqrt{4M}~\omega_{n}\). From (3.6) we have
$$\int_{R^3} F(x, \widetilde{{\omega_n}})dx \leq\xi\int_{R^3}|\widetilde{{\omega_n}}|^2 +C_{\xi} \int_{R^3} |\widetilde{{\omega _n}}|^q dx \rightarrow0, $$
(3.10)
as n→∞. Then, choosing n sufficiently large we conclude which means that \(\lim_{n\rightarrow\infty}\Phi_{\lambda_{n}}(t_{n}u_{n}) = \infty\). By the definition of tn, we see that \(\langle\Phi_{\lambda_{n}}' (t_{n}u_{n}),t_{n}u_{n}\rangle=0\). Consequently, by (f4)′ we have as n→∞. This leads to a contradiction and the conclusion follows. □

Proof of Theorem 1.2

Combining Lemmas 3.2, 3.3 and the compactness imbedding ELs(R3), 2≤s<2, standard arguments imply that there exists a convergent subsequence of {un} when λn→1. Notice that {un} is relevant to the choice of k, we may assume that unuk for some ukE. Moreover, from the equality
$$\Phi(u_n)=\Phi_{\lambda_n}(u_n)+(\lambda_n -1) \int_{R^3}F(x,u_n)dx, $$
(3.11)
and boundedness of {un} in E, we conclude that \(|\int_{R^{3}}F(x,u_{n})dx|<\infty\) as n→∞. Recalling that \(\Phi_{\lambda_{n}}(u_{n}) \in[\widetilde{{b_{k}}},\widetilde{{c_{k}}}],\lambda_{n}\rightarrow1\), we obtain
$$\Phi(u^k) = \lim_{n\rightarrow\infty} \Phi_{\lambda_n}(u_n) \in [\widetilde{b_k},\widetilde{c_k}]. $$
(3.12)
On the other hand, we have
$$\langle\Phi'(u_n),v\rangle=\langle\Phi_{\lambda_n}'(u_n),v\rangle+(\lambda_n -1) \int_{R^3}f(x,u_n)v dx$$
for all vE. Since \(\Phi_{\lambda_{n}}'(u_{n})=0,~\{u_{n}\}\) is bounded in E, the above equality yields that
$$\lim_{n\rightarrow\infty} \langle\Phi'(u_n),v\rangle=0 \quad\mbox{for all } v \in E. $$
(3.13)
In view of Φ∈C1(E,ℝ), we have Φ′(un)→Φ′(uk) in E. Therefore, for every vE,
$$|\langle\Phi'(u_n)-\Phi'(u^k),v\rangle|\leq\|\Phi'(u_n)-\Phi '(u^k)\|_{E^*}\|v\|_E\rightarrow0$$
as n→∞. This means that 〈Φ′(uk),v〉=0 for all vE, i.e. Φ′(uk)=0 in E. By (3.12) and \(\widetilde{b_{k}}\rightarrow+\infty\), we see that \(\{u^{k}\}_{k=1}^{\infty}\) is an unbounded sequence of critical points of functional Φ(u). This completes the proof. □

Remark 3.1

There is a nonlinearity f which satisfies the conditions of Theorem 1.2, but does not satisfy the conditions of Theorem 1.1, especially the Ambrosetti-Rabinowitz-type growth condition (f3). In fact, let f(x,s)=s3(4+ln(1+|s|)). Simple computation yields that and it is easy to check that f satisfies conditions (f1),(f2),(f3)′ and (f4)′.
But f does not satisfy (f3). Indeed, suppose by contradiction that there is some μ>4 such that μF(x,s)≤f(x,s)s for |s| large. That is, the following inequality holds true for |s| large. But this is impossible since μ>4.

Acknowledgements

The authors are grateful for the anonymous referees for very helpful suggestions and comments. This work was supported by NSFC Grants 10971238 and the Fundamental Research Funds for the Central Universities 0910KYZY51.

Copyright information

© Korean Society for Computational and Applied Mathematics 2012

Authors and Affiliations

  1. 1.College of ScienceMinzu University of ChinaBeijingP.R. China

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