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

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.

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 ${sup}_n\parallel u_n^k(\lambda )\parallel <\mathrm{\infty }$, ${\mathrm{\Phi }}_{\lambda }^{\prime }(u_n^k(\lambda ))\to 0$ and ${\mathrm{\Phi }}_{\lambda }(u_n^k(\lambda ))\to c_k(\lambda )$ as $n\to \mathrm{\infty }$.

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

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 $k\to \mathrm{\infty }$.

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).

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 $\lambda \in [1,2]$.

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

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

for all $u\in Y_k$ with $\parallel u\parallel \ge \frac{L_1}{{ϵ}_1}$. Since $\gamma <1$, 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 $\lambda \in [1,2]$.

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. □

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 $\lambda \in [1,2]$.

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 $b_k(\lambda ):={max}_{u\in Y_k,\parallel u\parallel =r_k}{J}_{\lambda }(u)<0$ for all $\lambda \in [1,2]$.

Since ${\zeta }_1<\gamma +1<2$, for $\parallel u\parallel =r_k<{\rho }_k$ small enough, we can get $J_{\lambda }(u)<0$ for all $\lambda \in [1,2]$. □

for some $M_1>0$. Since $1<\gamma +1<2$, (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. □