Infinitely many solutions for a fourth-order differential equation on a nonlinear elastic foundation

In this paper, existence results of infinitely many solutions for a fourth-order differential equation with nonlinear boundary conditions are established. The proof is based on variational methods. Some recent results are improved and extended.

In this paper, we consider a beam equation with nonlinear boundary conditions of the type

where f\in C([0,1],\mathbb{R}) and g\in C(\mathbb{R}) are real functions. This kind of problem arises in the study of deflections of elastic beams on nonlinear elastic foundations. The problem has the following physical description: a thin flexible elastic beam of length 1 is clamped at its left end x=0 and resting on an elastic device at its right end x=1, which is given by g. Then, the problem models the static equilibrium of the beam under a load, along its length, characterized by f. The derivation of the model can be found in [1, 2].

Owing to the importance of fourth-order two-point boundary value problems in describing a large class of elastic deflection, there is a wide literature that deals with the existence and multiplicity results for such a problem with different boundary conditions (see, for instance, [3–8] and the references therein).

Motivated by the above works, in the present paper we study the existence of infinitely many solutions for problem (1.1) when the nonlinear term f(x,u) satisfies the superlinear condition and sublinear condition at the infinity on u, respectively. As far as we know, this case has never before been considered.

Now we state our main results.

1.1 The superlinear case

We give the following assumptions.

• ({\mathcal{H}}_1) g is odd and satisfies

  2{\int }_0^{s}g(t)dt-g(s)s\ge 0,{\int }_0^{s}g(t)dt\ge 0\text{for all }s\in \mathbb{R}.

• ({\mathcal{H}}_2) There exist constants a,b\ge 0 and \gamma \in [0,1) such that

  |g(s)|\le a+b{|s|}^{\gamma }\text{for }s\in \mathbb{R}.

• ({\mathcal{H}}_3) {lim}_{|u|\to +\mathrm{\infty }}\frac{F(x,u)}{u^2}=+\mathrm{\infty } uniformly for x\in [0,1], where F(x,u)={\int }_0^{u}f(x,t)dt.

• ({\mathcal{H}}_4) F(x,0)\equiv 0, 0\le F(x,u)=o({|u|}^2) as |u|\to 0 uniformly for x\in [0,1].

• ({\mathcal{H}}_5) There exist constants \alpha >1, , c_1,c_2>0 and L>0 such that for every x\in [0,1] and u\in \mathbb{R} with |u|\ge L,

  f(x,u)u-2F(x,u)\ge c_1{|u|}^{\alpha },|f(x,u)|\le c_2{|u|}^{\beta }.

Theorem 1.1 Assume that ({\mathcal{H}}_1)-({\mathcal{H}}_5) hold and F is even in u. Then problem (1.1) has infinitely many solutions.

Remark 1.1 There exist some functions satisfying ({\mathcal{H}}_3)-({\mathcal{H}}_5), but not satisfying the well-known (AR)-condition,

for some \theta >2.

For example, take f(x,u)=2uln{(1+u^2)}^2+\frac{4u^3}{1+u^2}. Then F(x,u)={|u|}^{2}ln{(1+{|u|}^2)}^2. Obviously, ({\mathcal{H}}_3)-({\mathcal{H}}_4) are satisfied. Note that

and

for L being large enough, which implies ({\mathcal{H}}_5). However, it is easy to see that f does not satisfy (AR)-condition.

1.2 The sublinear case

We make the following assumptions.

• ({\mathcal{S}}_1) g is odd and satisfies g(s)s\ge 0 for any s\in \mathbb{R}.

• ({\mathcal{S}}_2) There exist constants b>0 and \gamma \in [0,1) such that

  |g(s)|\le b{|s|}^{\gamma }\text{for }s\in \mathbb{R}.

• ({\mathcal{S}}_3) F(x,0)\equiv 0 for any x\in [0,1].

• ({\mathcal{S}}_4) There are constants k_1>0 and {\zeta }_1\in [1,2) with such that

  F(x,u)\ge k_1{|u|}^{{\zeta }_1}\text{for any }(x,u)\in [0,1]\times \mathbb{R}.

• ({\mathcal{S}}_5) There exist constants k_2>0 and {\zeta }_2\in [1,2) such that

  |f(x,u)|\le k_2{|u|}^{{\zeta }_2-1}\text{for any }(x,u)\in [0,1]\times \mathbb{R}.

Theorem 1.2 Assume that ({\mathcal{S}}_1)-({\mathcal{S}}_5) hold and F is even in u. Then problem (1.1) has infinitely many solutions.

Remark 1.2 The condition ({\mathcal{S}}_1) implies that {\int }_0^{s}g(t)dt\ge 0.

The remainder of this paper is organized as follows. In Section 2, some preliminary results are presented. In Section 3, we give the proofs of our main results.

In this section, the following two theorems will be needed in our argument. Let E be a Banach space with the norm \parallel \cdot \parallel and E=\overline{{⨁}_{j\in \mathbb{N}}{X}_j} with for any j\in \mathbb{N}. Set Y_k={⨁}_{j=0}^k{X}_j, Z_k=\overline{{⨁}_{j=k}^{\mathrm{\infty }}{X}_j} and B_k=\{u\in Y_k:\parallel u\parallel \le {\rho }_k\}, N_k=\{u\in Z_k:\parallel u\parallel =r_k\} for {\rho }_k>r_k>0. Consider the C^1-functional {\mathrm{\Phi }}_{\lambda }:E\to \mathbb{R} defined by

Assume that:

• (C₁) {\mathrm{\Phi }}_{\lambda } maps bounded sets to bounded sets uniformly for \lambda \in [1,2]. Furthermore, {\mathrm{\Phi }}_{\lambda }(-u)={\mathrm{\Phi }}_{\lambda }(u) for all (\lambda ,u)\in [1,2]\times E.

• (C₂) B(u)\ge 0 for all u\in E; A(u)\to \mathrm{\infty } or B(u)\to \mathrm{\infty } as \parallel u\parallel \to \mathrm{\infty }; or

  {({\mathrm{C}}_2)}^{\prime } B(u)\le 0 for all u\in E; B(u)\to -\mathrm{\infty } as \parallel u\parallel \to \mathrm{\infty }.

For k\ge 2, define {\mathrm{\Gamma }}_k:=\{\gamma \in C(B_k,E):\gamma \text{ is odd};\gamma {|}_{\partial B_k}=id\},

c_k(\lambda ):={inf}_{\gamma \in {\mathrm{\Gamma }}_k}{max}_{u\in B_k}{\mathrm{\Phi }}_{\lambda }(\gamma (u)),

b_k(\lambda ):={inf}_{u\in Z_k,\parallel u\parallel =r_k}{\mathrm{\Phi }}_{\lambda }(u),

a_k(\lambda ):={max}_{u\in Y_k,\parallel u\parallel ={\rho }_k}{\mathrm{\Phi }}_{\lambda }(u).

Theorem 2.1 ([[9], Theorem 2.1])

Assume that (C₁) and (C₂) (or {({\mathrm{C}}_2)}^{\prime }) hold. If b_k(\lambda )>a_k(\lambda ) for all \lambda \in [1,2], then c_k(\lambda )\ge b_k(\lambda ) for all \lambda \in [1,2]. Moreover, for a.e. \lambda \in [1,2], there exists a sequence {\{u_n^k(\lambda )\}}_{n=1}^{\mathrm{\infty }} such that , {\mathrm{\Phi }}_{\lambda }^{\prime }(u_n^k(\lambda ))\to 0 and {\mathrm{\Phi }}_{\lambda }(u_n^k(\lambda ))\to c_k(\lambda ) as n\to \mathrm{\infty }.

Theorem 2.2 ([[9], Theorem 2.2])

Suppose that (C₁) holds. Furthermore, we assume that the following conditions hold:

• (D₁) B(u)\ge 0; B(u)\to \mathrm{\infty } as \parallel u\parallel \to \mathrm{\infty } on any finite dimensional subspace of E.

• (D₂) There exist {\rho }_k>r_k>0 such that a_k(\lambda ):={inf}_{u\in Z_k,\parallel u\parallel ={\rho }_k}{\mathrm{\Phi }}_{\lambda }(u)\ge 0>b_k(\lambda ):={max}_{u\in Y_k,\parallel u\parallel =r_k}{\mathrm{\Phi }}_{\lambda }(u) for all \lambda \in [1,2] and d_k(\lambda ):={inf}_{u\in Z_k,\parallel u\parallel \le {\rho }_k}{\mathrm{\Phi }}_{\lambda }(u)\to 0 as k\to \mathrm{\infty } uniformly for \lambda \in [1,2].

Then there exist {\lambda }_n\to 1, u({\lambda }_n)\in Y_n such that {\mathrm{\Phi }}_{{\lambda }_n}^{\prime }{|}_{Y_n}(u({\lambda }_n))=0, {\mathrm{\Phi }}_{{\lambda }_n}(u({\lambda }_n))\to c_k\in [d_k(2),b_k(1)] as n\to \mathrm{\infty }. In particular, if \{u({\lambda }_n)\} has a convergent subsequence for every k, then {\mathrm{\Phi }}_1 has infinitely many nontrivial critical points \{u_k\}\subset E\setminus \{0\} satisfying {\mathrm{\Phi }}_1(u_k)\to 0^{-} as k\to \mathrm{\infty }.

Now we begin describing the variational formulation of problem (1.1), which is based on the function space

where H^2(0,1) is the Sobolev space of all functions u:[0,1]\to \mathbb{R} such that u and its distributional derivative u^{\prime } are absolutely continuous and u^{″} belongs to L^2(0,1). Then E is a Hilbert space equipped with the inner product and the norm

where {\parallel \cdot \parallel }_p denotes the standard L^p norm. In addition, E is compactly embedded in the spaces L^2(0,1) and C[0,1], and therefore, there exist immersion constants S_2,\overline{S}>0 such that

Next, we consider the functional J:E\to \mathbb{R} defined by

where F, G are the primitives

Since f, g are continuous, we deduce that J is of class C^1 and its derivative is given by

for all u,\phi \in E. Then we can infer that u\in E is a critical point of J if and only if it is a (classical) solution of problem (1.1).

Now we define a class of functionals on E by

It is easy to know that J_{\lambda }\in C^1(E;\mathbb{R}) for all \lambda \in [1,2] and the critical points of J_1=J correspond to the weak solutions of problem (1.1). We choose a completely orthonormal basis \{e_j\} of E and define X_j:=\mathbb{R}{e}_j. Then Z_k, Y_k can be defined as that at the beginning of Section 2.

We will prove Theorem 1.1 by using Theorem 2.1. Firstly, we give the following three useful lemmas.

Lemma 3.1 Under the assumptions of Theorem 1.1, there exists {\rho }_k>0 large enough such that a_k(\lambda ):={max}_{u\in Y_k,\parallel u\parallel ={\rho }_k}{J}_{\lambda }(u)\le 0 for all \lambda \in [1,2].

Proof Let u\in Y_k, then there exists {ϵ}_1>0 such that

Otherwise, for any positive integer n, there exists u_n\in Y_k\setminus \{0\} such that

for all k. Set v_n(x):=\frac{u_n(x)}{\parallel u_n\parallel }\in Y_k\setminus \{0\}, then \parallel v_n\parallel =1 and

for all k. Since , it follows from the compactness of the unit sphere of Y_k that there exists a subsequence, say \{v_n\}, such that v_n converges to some v_0 in Y_k. Hence, we have \parallel v_0\parallel =1. By the equivalence of the norms on the finite-dimensional space Y_k, we have v_n\to v_0 in L^2[0,1], i.e.,

Thus there exist {\xi }_1,{\xi }_2>0 such that

In fact, if not, we have

for all positive integer n. This implies that

as n\to \mathrm{\infty }, which gives a contradiction. Therefore, (3.4) holds.

Now let

and {\mathrm{\Omega }}_n^{\mathrm{\perp }}=[0,1]\setminus {\mathrm{\Omega }}_n. By (3.2) and (3.4), we have

for all positive integer n. Let n be large enough such that {\xi }_2-\frac{1}{n}\ge \frac{1}{2}{\xi }_2 and {\xi }_1-\frac{1}{n}\ge \frac{1}{2}{\xi }_1. Then we have

This implies that

for all large n, which is a contradiction with (3.3). Therefore, (3.1) holds.

For any u\in Y_k, let {\mathrm{\Omega }}_u=\{x\in [0,1]:|u(x)|\ge {ϵ}_1\parallel u\parallel \}. By condition ({\mathcal{H}}_3), for M=\frac{1}{\lambda {ϵ}_1^2}\ge \frac{1}{2{ϵ}_1^2}>0, there exists L_1>0 such that

Hence one has

for all u\in Y_k with \parallel u\parallel \ge \frac{L_1}{{ϵ}_1}. It follows from ({\mathcal{H}}_2)-({\mathcal{H}}_4) and (3.1) that

for all u\in Y_k with \parallel u\parallel \ge \frac{L_1}{{ϵ}_1}. Since , for \parallel u\parallel ={\rho }_k large enough, we have J_{\lambda }(u)\le 0. □

Lemma 3.2 Under the assumptions of Theorem 1.1, there exist r_k>0, \widetilde{b_k}\to \mathrm{\infty } such that b_k(\lambda ):={inf}_{u\in Z_k,\parallel u\parallel =r_k}{J}_{\lambda }(u)\ge \widetilde{b_k} for all \lambda \in [1,2].

Proof Set {\gamma }_k:={sup}_{u\in Z_k,\parallel u\parallel =1}{\parallel u\parallel }_{\mathrm{\infty }}. Then {\gamma }_k\to 0 as k\to \mathrm{\infty }. Indeed, it is clear that , so that {\gamma }_k\to \overline{\gamma }\ge 0, as k\to \mathrm{\infty }. For every k\ge 0, there exists u_k\in Z_k such that \parallel u_k\parallel =1 and {\parallel u_k\parallel }_{\mathrm{\infty }}>{\gamma }_k/2. By the definition of Z_k, u_k\rightharpoonup 0 in E. Then it implies that u_k\to 0 in C[0,1]. Thus we have proved that \overline{\gamma }=0. By ({\mathcal{H}}_5), we have

By ({\mathcal{H}}_4), for any ϵ>0, there exists \delta >0 such that

Therefore, there exists C=C(ϵ)>0 such that

Hence, for any u\in Z_k, choose ϵ={(4\lambda p^2)}^{-1}, by ({\mathcal{H}}_1) and (3.5), we have

Let r_k:={(8\lambda C{\gamma }_k^{\beta +1})}^{\frac{1}{1-\beta }}. Then, for any u\in Z_k with \parallel u\parallel =r_k, we have

uniformly for λ as k\to \mathrm{\infty }. □

Lemma 3.3 Under the assumptions of Theorem 1.1, there exist {\lambda }_n\to 1 as n\to \mathrm{\infty }, {\{u_n(k)\}}_{n=1}^{\mathrm{\infty }}\subset E such that J_{{\lambda }_n}^{\prime }(u_n(k))\to 0, J_{{\lambda }_n}(u_n(k))\in [\widetilde{b_k},\widetilde{c_k}], where \widetilde{c_k}={sup}_{u\in B_k}{\mathrm{\Phi }}_1(u).

Proof It is easy to verify that (C₁) and (C₂) of Theorem 2.1 hold. By Lemmas 3.1, 3.2 and Theorem 2.1, we can obtain the result. □

Proof of Theorem 1.1 For the sake of notational simplicity, in what follows we always set u_n=u_n(k) for all n\in \mathbb{N}. By Lemma 3.3, it suffices to prove that {\{u_n\}}_{n=1}^{\mathrm{\infty }} is bounded and possesses a strong convergent subsequence in E. If not, passing to a subsequence if necessary, we assume that \parallel u_n\parallel \to \mathrm{\infty } as n\to \mathrm{\infty }. In view of ({\mathcal{H}}_5), there exists c_5>0 such that

and combining ({\mathcal{H}}_1), we have

This implies that

Note that from ({\mathcal{H}}_5), . Let \eta =\frac{\alpha -1}{\alpha (\beta -1)}, then

By ({\mathcal{H}}_5), there exists c_6>0 such that

By (2.5), ({\mathcal{H}}_2) and the Hölder inequality, one has

where C_{\eta }>0 is a constant independent of n. By (3.8) we obtain

combining this inequality with (3.6) and (3.7) yields that

as n\to \mathrm{\infty }. Combining this with (3.9), we have

since . This is a contradiction. Therefore, {\{u_n\}}_{n=1}^{\mathrm{\infty }} is bounded in E. Without loss of generality, we may assume u_n\rightharpoonup w_k in E. Then u_n\to w_k in C[0,1]. Note that

Taking n\to \mathrm{\infty }, we have {lim}_{n\to \mathrm{\infty }}\parallel u_n-w_k\parallel =0, which means that u_n\to w_k in E and J_1^{\prime }(w_k)=0. Hence, J_1 has a critical point w_k with J_1(w_k)\in [\widetilde{b_k},\widetilde{c_k}]. Consequently, we obtain infinitely many solutions since \widetilde{b_k}\to \mathrm{\infty }. □

Lemma 3.4 Under the assumptions of Theorem 1.2, there exists {\rho }_k small enough such that a_k(\lambda ):={inf}_{u\in Z_k,\parallel u\parallel ={\rho }_k}{J}_{\lambda }(u)\ge 0 and d_k(\lambda ):={inf}_{u\in Z_k,\parallel u\parallel \le {\rho }_k}{J}_{\lambda }(u)\to 0 as k\to \mathrm{\infty } uniformly for \lambda \in [1,2].

Proof For any u\in Z_k, by using {\gamma }_k:={sup}_{u\in Z_k,\parallel u\parallel =1}{\parallel u\parallel }_{\mathrm{\infty }} defined in Lemma 3.2, together with ({\mathcal{S}}_1) and ({\mathcal{S}}_5), we have

for all u\in Z_k with \parallel u\parallel ={\rho }_k:={(4\lambda k_2{\gamma }_k^{{\zeta }_2})}^{1/(2-{\zeta }_2)}. Obviously, {\rho }_k\to 0 as k\to \mathrm{\infty }. So a_k(\lambda ):={inf}_{u\in Z_k,\parallel u\parallel ={\rho }_k}{J}_{\lambda }(u)\ge 0 and d_k(\lambda ):={inf}_{u\in Z_k,\parallel u\parallel \le {\rho }_k}{J}_{\lambda }(u)\to 0 as k\to \mathrm{\infty } uniformly for \lambda \in [1,2]. □

Lemma 3.5 Under the assumptions of Theorem 1.2, there exists r_k small enough such that for all \lambda \in [1,2].

Proof For any u\in Y_k, by ({\mathcal{S}}_1)-({\mathcal{S}}_5) and the equivalence of the norms on the finite-dimensional space Y_k, we have

Since , for small enough, we can get for all \lambda \in [1,2]. □

Proof of Theorem 1.2 It is easy to verify that (C₁) and (D₁) hold under the assumptions of Theorem 1.2. By Lemmas 3.4 and 3.5, the condition (D₂) is also satisfied. Therefore, by Theorem 2.2 there exist {\lambda }_n\to 1, u({\lambda }_n):=u_n\in Y_n such that J_{{\lambda }_n}^{\prime }{|}_{Y_n}(u_n)=0, J_{{\lambda }_n}(u_n)\to c_k\in [d_k(2),b_k(1)] as n\to \mathrm{\infty }. In the following we show that {\{u_n\}}_{n=1}^{\mathrm{\infty }} is bounded. Indeed, note that

for some M_1>0. Since , (3.10) yields that \{u_n\} is bounded in E. By a standard argument, this yields a critical point u^k of J_1 such that J_1(u^k)\in [d_k(2),c_k(1)]. Since d_k(2)\to 0^{-} as k\to \mathrm{\infty }, we can obtain infinitely many critical points. □

The author would like to express her sincere thanks to the referees for their helpful comments.

Affiliations

1. Business School, Shandong University of Technology, Zibo, 255049, P.R. China
   • Xiaodan Wang

Corresponding author

Correspondence to Xiaodan Wang.

Competing interests

The author declares that she has no competing interests.

Authors’ contributions

The author read and approved the final manuscript.

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