On the multiplicity of solutions for a kind of fourth-order equation depending on two real parameters

Abstract

In this paper, by suitable assumptions on nonlinear boundary term, we establish the existence of three distinct weak solutions for a kind of fourth-order boundary value problem depending on two parameters.

1 Introduction

In this paper, we consider the following fourth-order problem:

$$ \textstyle\begin{cases} u^{(iv)}(x)= \lambda f(x,u(x)), \quad \text{in } [0,1], \\ u(0)=u'(0)=0, \\ u''(1)=0,\qquad u'''(1)=\mu g(u(1)), \end{cases} $$

(1.1)

where $\lambda , \mu \in {]0 , \infty [}$, $g:\mathbb{R}\rightarrow \mathbb{R}$ is a continuous function and $f:[0,1]\times \mathbb{R}\rightarrow \mathbb{R}$ is an $L^{1}$-Caratéodory function.

Problem (1.1) describes the static equilibrium of a flexible elastic beam of length 1 when, along its length, a load f is added to cause deformation. Precisely, conditions $u(0)=u'(0)=0$ mean that the left end of the beam is fixed and conditions $u''(1)=0$, $u'''(1)=\mu g(u(1))$ mean that the right end of the beam is attached to a bearing device, given by the function g.

Existence and multiplicity of solutions for fourth-order boundary value problems have been discussed by several authors in the last decades; see for example [1, 4, 5, 9–13, 16, 17, 19–21] and the references therein.

In particular, Yang et al. [21] used Ricceri’s variational principle [18] to establish the existence of at least two classical solutions generated from g for problem (1.1) with $\mu =1$.

The authors in [9], using a multiplicity result by Cabada and Iannizzotto [8], ensured the existence of at least two nontrivial classical solutions for the problem

$$ \textstyle\begin{cases} u^{(4)}(x)+\lambda f(x,u(x))=0, \quad 0< x< 1, \\ u(0)=u'(0)=u''(1)=0, \\ u'''(1)=\lambda g(u(1)), \end{cases} $$

where the functions $f:[0,1]\times \mathbb{R}\to \mathbb{R}$ and $g:\mathbb{R}\to \mathbb{R}$ are continuous and $\lambda \geq 0$ is a real parameter.

Bonanno et al. [4], by means of an abstract critical points result of Bonanno [2], studied the existence of at least one nonzero classical solution for problem (1.1).

In [12], by using a smooth version of [7, Theorem 2.1], Heidarkhani et al. established the existence of infinitely many generalized solutions for the following perturbed fourth-order problem:

$$ \textstyle\begin{cases} u^{(4)}(t)= \lambda f(t,u(t))+\mu g(t,u(t))+p(u(t)), \quad 0< t< 1, \\ u(0)=u'(0)=0, \\ u''(1)=0,\qquad u'''(1)=h(u(1)), \end{cases} $$

where $\lambda >0$, $\mu \ge 0$ are two parameters, f, g are two $L^{2}$-Caratéodory functions, and p, h are Lipschitz continuous functions such that $p(0)=h(0) = 0$.

Also in [11], the present authors obtained sufficient conditions to guarantee that problem (1.1) has infinitely many classical solutions.

More recently, Heidarkhani and Gharehgazlouei [13], using an immediate consequence of [3, Theorem 3.3], ensured the existence of at least three generalized solutions for the problem

$$ \textstyle\begin{cases} u^{(iv)}(x)= \lambda f(x,u(x))+h(u(x)), \quad \text{in } [0,1], \\ u(0)=u'(0)=u''(1)=0, \\ u'''(1)=\mu g(u(1)), \end{cases} $$

where $\lambda >0$, $\mu \ge 0$ are two parameters, $f:[0,1]\times \mathbb{R}\rightarrow \mathbb{R}$ is an $L^{1}$-Caratéodory function, $g:\mathbb{R}\rightarrow \mathbb{R}$ is a nonnegative continuous function, and $h:\mathbb{R}\rightarrow \mathbb{R}$ is a Lipschitz continuous function such that $h(0) = 0$.

Motivated by the above works, the aim of the present paper is to offer the existence of three solutions for fourth-order problem (1.1) by using two kinds of critical point theorems obtained in [3, 6].

For completeness, we cite the recent and nice works [14, 15] as general references on the subject treated in this paper.

2 Abstract setting

In order to study problem (1.1), the variational setting is the space

$$ X:=\bigl\{ u\in H^{2}\bigl([0,1]\bigr) : u(0)=u'(0)=0 \bigr\} , $$

where $H^{2}([0,1])$ is the Sobolev space of all function $u:[0,1]\rightarrow \mathbb{R}$ such that u and its distributional derivative $u'$ are absolutely continuous and $u''$ belongs to $L^{2}([0,1])$. X is a Hilbert space with the inner product

$$ \langle u , v\rangle := \int _{0}^{1} u''(t)v''(t)\,dt $$

and the corresponding norm

$$ \Vert u \Vert := \biggl( \int _{0}^{1} \bigl(u''(t) \bigr)^{2} \,dt \biggr)^{\frac{1}{2}}. $$

We observe that the norm $\|\cdot \|$ on X is equivalent to the usual norm

$$ \int _{0}^{1} \bigl( \bigl\vert u(t) \bigr\vert ^{2}+ \bigl\vert u'(t) \bigr\vert ^{2}+ \bigl\vert u''(t) \bigr\vert ^{2} \bigr)\,dt. $$

It is well known that the embedding $X\hookrightarrow C^{1}([0,1])$ is compact and

$$ \Vert u \Vert _{C^{1}([0,1])} := \max \bigl\{ \Vert u \Vert _{\infty }, \bigl\Vert u' \bigr\Vert _{\infty } \bigr\} \leq \Vert u \Vert $$

(2.1)

for all $u\in X$ (see [21]).

We say that $u\in X$ is a weak solution of problem (1.1) whenever

$$ \int _{0}^{1} u''(x)v''(x)\,dx -\lambda \int _{0}^{1} f\bigl(x,u(x)\bigr)v(x)\,dx+ \mu g \bigl(u(1)\bigr)v(1)=0 $$

for all $v\in X$. By a classical solution of problem (1.1) we mean a function $u\in C^{1}([0,1])$ such that $u^{(iv)}(x)\in C([0,1])$ and the boundary conditions and the equation are satisfied in $[0,1]$.

Our main tools are critical point theorems that we recall here in a convenient form. The first result has been obtained in [6], and it is a more precise version of Theorem 3.2 of [3]. The second one has been established in [3].

Lemma 2.1

([6, Theorem 3.6])

Let X be a reflexive real Banach space; $\Phi :X\rightarrow \mathbb{R}$ be a coercive, continuously Gâteaux differentiable, and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on $X^{*}$; $\Psi :X\rightarrow \mathbb{R}$ be a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact such that

$$ \Phi (0)=\Psi (0)=0. $$

Assume that there exist $r>0$ and $\overline{x}\in X$, with $r<\Phi (\overline{x})$, such that

$\mathrm{(a_{1})}$:

$\frac{\sup_{\Phi (x)\leq r}\Psi (x)}{r}< \frac{\Psi (\overline{x})}{\Phi (\overline{x})}$;

$\mathrm{(a_{2})}$:

for each $\lambda \in \Lambda _{r}:={] \frac{\Phi (\overline{x})}{\Psi (\overline{x})}, \frac{r}{\sup_{\Phi (x)\leq r}\Psi (x)}[}$, the functional $\Phi -\lambda \Psi$ is coercive.

Then, for each $\lambda \in \Lambda _{r}$, the functional $\Phi -\lambda \Psi$ has at least three distinct critical points in X.

Lemma 2.2

([3, Theorem 3.3])

Let X be a reflexive real Banach space; $\Phi :X\rightarrow \mathbb{R}$ be a convex, coercive, and continuously Gâteaux differentiable functional whose derivative admits a continuous inverse on $X^{\ast }$; $\Psi :X\rightarrow \mathbb{R}$ be a continuously Gâteaux differentiable functional whose derivative is compact such that

$$ \inf_{X}\Phi =\Phi (0)=\Psi (0)=0. $$

Assume that there are two positive constants $r_{1}$, $r_{2}$ and $\overline{x}\in X$, with $2r_{1}<\Phi (\overline{x})<\frac{r_{2}}{2}$, such that

$\mathrm{(b_{1})}$:

$\frac{\sup_{\Phi (x)< r_{1}}\Psi (x)}{r_{1}}< \frac{2}{3}\frac{\Psi (\overline{x})}{\Phi (\overline{x})}$;

$\mathrm{(b_{2})}$:

$\frac{\sup_{\Phi (x)< r_{2}}\Psi (x)}{r_{2}}< \frac{1}{3}\frac{\Psi (\overline{x})}{\Phi (\overline{x})}$;

$\mathrm{(b_{3})}$:

for each λ in

$$ \Lambda ' := {\biggl]\frac{3}{2} \frac{\Phi (\overline{x})}{\Psi (\overline{x})}, \min \biggl\{ \frac{r_{1}}{\sup_{\Phi (x)< r_{1}}\Psi (x)}, \frac{\frac{r_{2}}{2}}{\sup_{\Phi (x)< r_{2}}\Psi (x)} \biggr\} \biggr[} $$

and for every $x_{1},x_{2}\in X$, which are local minima for the functional $\Phi -\lambda \Psi$ and such that $\Psi (x_{1})\geq 0$ and $\Psi (x_{2})\geq 0$, one has $\inf_{s\in [0,1]}\Psi (sx_{1}+(1-s)x_{2})\geq 0$.

Then, for each $\lambda \in \Lambda '$, the functional $\Phi -\lambda \Psi$ has at least three distinct critical points which lie in $\Phi ^{-1}(]-\infty ,r_{2}[)$.

We use the following notations:

Corresponding to f, g, we introduce the functions F, G as follows:

$$ F(x,\xi ):= \int _{0}^{\xi }f(x,t)\,dt,\qquad G(\xi ):=- \int _{0}^{\xi }g(t)\,dt $$

for all $x\in [0, 1]$ and $\xi \in \mathbb{R}$. Also, for each θ and η of positive real numbers, define

$$ F^{\theta }= \int _{0}^{1} \sup_{|\xi |\leq \theta } F(x,\xi ) \,dx, \qquad G^{\theta }=\sup_{|\xi |\leq \theta } G(\xi ),\qquad G_{\eta }=\inf_{|\xi |\leq \eta } G(\xi ). $$

3 Main results

In this section, we present our main result on the existence of at least three weak solutions for problem (1.1).

In order to introduce our result, we fix $\theta ,\eta >0$ such that

$$ \frac{32\pi ^{4} \eta ^{2}}{27\int _{\frac{3}{4}}^{1} F(x,\eta ) \,dx}< \frac{\theta ^{2}}{2F^{\theta }}, $$

and pick

$$ \Lambda := {\biggl] \frac{32\pi ^{4} \eta ^{2}}{27\int _{\frac{3}{4}}^{1} F(x,\eta ) \,dx}, \frac{\theta ^{2}}{2F^{\theta }} \biggr[}. $$

(3.1)

Set

$$ \delta := \min \biggl\{ \frac{\theta ^{2}-2\lambda F^{\theta }}{2 G^{\theta }}, \frac{\frac{32}{27} \pi ^{4}\eta ^{2} -\lambda \int _{\frac{3}{4}}^{1} F(x,\eta ) \,dx}{ G(\eta )} \biggr\} $$

(3.2)

and

$$ \overline{\delta }:=\min \biggl\{ \delta , \frac{1}{\max \{ 0,4\limsup_{|\xi |\rightarrow +\infty }\frac{G(\xi )}{\xi ^{2}} \} } \biggr\} , $$

(3.3)

where we read ${r}/{0}=+\infty$. For instance, $\overline{\delta }=+\infty$ when $\limsup_{|\xi |\rightarrow +\infty }\frac{G(\xi )}{\xi ^{2}} \leq 0$ and $G(\eta )=G^{\theta }=0$.

With the above notations we are able to prove the following multiplicity property.

Theorem 3.1

Assume that there exist two positive constants θ, η, with $\theta <\frac{8}{3\sqrt{3}}\pi ^{2} \eta$, such that

$\mathrm{(A1)}$:

$F(x,t) \geq 0$ for each $(x,t)\in [ \frac{3}{8} , \frac{3}{4}]\times [0,\eta ]$;

$\mathrm{(A2)}$:

$\frac{F^{\theta }}{\theta ^{2}}< \frac{27}{64 \pi ^{4}} \frac{\int _{\frac{3}{4}}^{1} F(x,\eta ) \,dx}{\eta ^{2}}$;

$\mathrm{(A3)}$:

$\limsup_{|\xi |\rightarrow +\infty } \frac{\sup_{x\in [0,1]}F(x,\xi )}{\xi ^{2}}< \frac{F^{\theta }}{2\theta ^{2}}$.

Then, for every $\lambda \in \Lambda$, where Λ is given by (3.1), and for every continuous function $g:\mathbb{R}\rightarrow \mathbb{R}$ such that

$$ \limsup_{|\xi |\rightarrow +\infty } \frac{G(\xi )}{\xi ^{2}}< + \infty , $$

there exists $\overline{\delta }>0$ given by (3.3) such that, for each $\mu \in [0,\overline{\delta }[$, problem (1.1) admits at least three distinct weak solutions.

Proof

Our aim is to apply Lemma 2.1 to problem (1.1). To this end, we introduce the functionals $\Phi , \Psi : X \rightarrow \mathbb{R}$ as follows:

$$\begin{aligned}& \Phi (u):=\frac{1}{2} \Vert u \Vert ^{2}, \\& \Psi (u):= \int _{0}^{1} F\bigl(x,u(x)\bigr)\,dx+ \frac{{\mu }}{{\lambda }}G\bigl(u(1)\bigr). \end{aligned}$$

It is well known that Ψ is a differentiable functional whose differential at the point $u\in X$ is

$$ \Psi '(u) (v)= \int _{0}^{1} f\bigl(x,u(x)\bigr)v(x)\,dx- \frac{{\mu }}{{\lambda }} g\bigl(u(1)\bigr)v(1) $$

for every $v\in X$. Moreover, in [21], the authors proved that $\Psi '$ is strongly continuous on X, which implies that $\Psi '$ is a compact operator. Furthermore, by standard arguments, Φ is coercive and continuously differentiable whose differential at the point $u\in X$ is

$$ \Phi '(u) (v)= \int _{0}^{1}u''(x)v''(x)\,dx $$

for each $v\in X$. Also, in [21] it is proved that $\Phi '$ admits a continuous inverse on $X^{*}$. Moreover, Φ is sequentially weakly lower semicontinuous. One can show that the weak solutions of problem (1.1) are exactly the solutions of the equation $\Phi '(u)- \lambda \Psi '(u)=0$.

Fix $\lambda \in \Lambda$ and put $r=\frac{\theta ^{2}}{2}$. Then, for $u\in X$ with $\Phi (u) \leq r$,

$$\begin{aligned} \sup_{\Phi (u)\leq r}\Psi (u) =& \sup_{\Phi (u)\leq r} \biggl( \int _{0}^{1} F\bigl(x,u(x)\bigr) \,dx + \frac{\mu }{\lambda } G\bigl(u(1)\bigr) \biggr) \\ \leq & \int _{0}^{1} \sup_{|t|\leq \theta }F(x,t) \,dx+ \frac{\mu }{\lambda } G^{\theta }= F^{\theta }+\frac{\mu }{\lambda } G^{\theta }. \end{aligned}$$

Therefore,

$$ \frac{\sup_{\Phi (u)\leq r}\Psi (u)}{r} \leq \frac{2F^{\theta }}{\theta ^{2}}+ \frac{2\mu }{\lambda } \frac{G^{\theta }}{\theta ^{2}}. $$

From this, if $G^{\theta }=0$, it is clear that

$$ \frac{\sup_{\Phi (u)\leq r}\Psi (u)}{r}< \frac{1}{\lambda }, $$

(3.4)

while if $G^{\theta }>0$, it turns out to be true bearing in mind that

$$ \mu < \frac{\theta ^{2}-2\lambda F^{\theta }}{2G^{\theta }}. $$

Denote by w the function of X defined by

$$ w(x):=\textstyle\begin{cases} 0,& x\in [0,\frac{3}{8}], \\ \eta \cos ^{2} (\frac{4\pi x}{3} ),& x\in {] \frac{3}{8} , \frac{3}{4}[}, \\ \eta , & x\in [ \frac{3}{4}, 1]. \end{cases} $$

(3.5)

It is easy to see that

$$ \Phi (w)=\frac{32}{27}\pi ^{4}\eta ^{2}. $$

Therefore, since $\theta <\frac{8}{3\sqrt{3}}\pi ^{2}\eta$, one has $\Phi (w)>r$.

From assumption $\mathrm{(A1)}$ we obtain

$$ \Psi (w)= \int _{0}^{1} F\bigl(x,w(x)\bigr) \,dx+ \frac{\mu }{\lambda } G\bigl(w(1)\bigr)\geq \int _{\frac{3}{4}}^{1} F(x,\eta )\,dx+\frac{\mu }{\lambda }G(\eta ). $$

So, we have

$$ \frac{\Psi (w)}{\Phi (w)}\geq \frac{\int _{\frac{3}{4}}^{1} F(x,\eta )\,dx+\frac{\mu }{\lambda }G(\eta )}{\frac{32}{27}\pi ^{4}\eta ^{2}}. $$

Hence, if $G(\eta )\geq 0$, we find

$$ \frac{\Psi (w)}{\Phi (w)}>\frac{1}{\lambda }, $$

(3.6)

while if $G(\eta ) < 0$, the same relation holds since

$$ \mu G(\eta ) > \frac{32}{27}\pi ^{4}\eta ^{2} - \lambda \int _{ \frac{3}{4}}^{1} F(x,\eta ) \,dx. $$

Now, taking into account (3.4) and (3.6) results in

$$ \frac{\Psi (w)}{\Phi (w)} > \frac{1}{\lambda } > \frac{\sup_{\Phi (u)\leq r} \Psi (u)}{r}, $$

and condition $\mathrm{(a_{1})}$ of Lemma 2.1 is verified.

Now, in order to prove the coercivity of the functional $\Phi -\lambda \Psi$, first we assume that

$$ \limsup_{|\xi |\rightarrow +\infty } \frac{\sup_{x\in [0,1]}F(x,\xi )}{\xi ^{2}}>0. $$

Therefore, fix

$$ \limsup_{|\xi |\rightarrow +\infty } \frac{\sup_{x\in [0,1]}F(x,\xi )}{\xi ^{2}}< \varepsilon < \frac{F^{\theta }}{2\theta ^{2}}, $$

from $\mathrm{(A3)}$ there is a function $h_{\varepsilon }\in L^{1}([0,1])$ such that

$$ F(x,\xi )\leq \varepsilon \xi ^{2}+h_{\varepsilon }(x) $$

for each $x\in [0,1]$ and $\xi \in \mathbb{R}$. Taking (2.1) into account and since $\lambda <\frac{\theta ^{2}}{2F^{\theta }}$, it follows that

$$\begin{aligned} \lambda \int _{0}^{1} F\bigl(x,u(x)\bigr) \,dx \leq &\lambda \biggl(\varepsilon \int _{0}^{1}\bigl(u(x)\bigr)^{2} \,dx + \int _{0}^{1} h_{\varepsilon }(x) \,dx \biggr) \\ < &\frac{\theta ^{2}}{2F^{\theta }} \biggl(\varepsilon \int _{0}^{1}\bigl(u(x)\bigr)^{2} \,dx+ \int _{0}^{1} h_{\varepsilon }(x) \,dx \biggr) \\ \leq &\frac{\theta ^{2}}{2F^{\theta }} \bigl(\varepsilon \Vert u \Vert ^{2}+ \Vert h_{\varepsilon } \Vert _{L^{1}([0,1])} \bigr) \end{aligned}$$

(3.7)

for each $u\in X$. Moreover, since $\mu <\overline{\delta }$, we obtain

$$ \limsup_{|\xi |\rightarrow +\infty }\frac{G(\xi )}{\xi ^{2}}< \frac{1}{4\mu } . $$

Thus, there is a positive constant $h_{\mu }$ such that

$$ G(\xi )\leq \frac{1}{4\mu }\xi ^{2}+h_{\mu } $$

for each $\xi \in \mathbb{R}$. Thus, taking again (2.1) into account, it follows that

$$\begin{aligned} G\bigl(u(1)\bigr) \leq &\frac{1}{4\mu }\bigl(u(1) \bigr)^{2}+h_{\mu } \\ \leq &\frac{1}{4\mu } \Vert u \Vert ^{2}+h_{\mu } \end{aligned}$$

(3.8)

for each $u\in X$. Finally, putting together (3.7) and (3.8), we have

$$\begin{aligned} \Phi (u)-\lambda \Psi (u) \geq &\frac{1}{2} \Vert u \Vert ^{2}- \frac{\theta ^{2}}{ 2F^{\theta }} \bigl(\varepsilon \Vert u \Vert ^{2}+ \Vert h_{\varepsilon } \Vert _{L^{1}([0,1])} \bigr) - \frac{1}{4} \Vert u \Vert ^{2}-\mu h_{\mu } \\ =&\frac{1}{2} \biggl(\frac{1}{2}-\frac{\theta ^{2}}{F^{\theta }} \varepsilon \biggr) \Vert u \Vert ^{2} - \frac{\theta ^{2} \Vert h_{\varepsilon } \Vert _{L^{1}([0,1])}}{2F^{\theta }}-\mu h_{\mu }. \end{aligned}$$

On the other hand, if

$$ \limsup_{|\xi |\rightarrow +\infty } \frac{\sup_{x\in [0,1]}F(x,\xi )}{\xi ^{2}}\leq 0, $$

there exists a function $h_{\varepsilon }\in L^{1}([0,1])$ such that $F(x,\xi )\leq h_{\varepsilon }(x)$ for each $x\in [0,1]$ and $\xi \in \mathbb{R}$, and arguing as before we obtain

$$ \Phi (u)-\lambda \Psi (u)\geq \frac{1}{4} \Vert u \Vert ^{2}- \frac{\theta ^{2} \Vert h_{\varepsilon } \Vert _{L^{1}([0,1])}}{2F^{\theta }}-\mu h_{\mu }. $$

Both cases lead to the coercivity of $\Phi -\lambda \Psi$, and condition $\mathrm{(a_{2})}$ of Lemma 2.1 is verified.

Since from (3.4) and (3.6)

$$ \lambda \in \Lambda \subseteq {\biggl]\frac{\Phi (w)}{\Psi (w)}, \frac{r}{\sup_{\Phi (u)\leq r}\Psi (u)} \biggr[}, $$

Lemma 2.1 ensures the existence of at least three critical points for the functional $\Phi -\lambda \Psi$, and the proof is complete. □

Theorem 3.2

Let $\theta _{1}$, $\theta _{2}$, and η be positive constants such that $\frac{3}{4}\sqrt{\frac{3}{2}}\frac{\theta _{1}}{\pi ^{2}} < \eta < \frac{3}{8}\sqrt{\frac{3}{2}}\frac{\theta _{2}}{\pi ^{2}}$ and $f:[0 , 1] \times \mathbb{R}\rightarrow \mathbb{R}$ be a mapping for which $f(x,t)\geq 0$ for every $(x,t)\in [0 , 1] \times [0 , \theta _{2}]$. Assume that

• (B1)
  $$ \frac{\int _{0}^{1} F(x,\theta _{1}) \,dx}{\theta _{1}^{2}} < \frac{64}{81} \frac{\pi ^{4} \int _{\frac{3}{4}}^{1} F(x,\eta ) \,dx}{\eta ^{2}}; $$
• (B2)
  $$ \frac{\int _{0}^{1} F(x,\theta _{2}) \,dx}{\theta _{2}^{2}} < \frac{32}{81} \frac{\pi ^{4} \int _{\frac{3}{4}}^{1} F(x,\eta ) \,dx}{\eta ^{2}}. $$

Then, for each

$$ \lambda \in \Lambda ':= {\biggl] \frac{32}{18} \frac{\pi ^{4} \eta ^{2}}{\int _{\frac{3}{4}}^{1} F(x,\eta ) \,dx}, \frac{1}{2} \min \biggl\{ \frac{\theta _{1}^{2}}{\int _{0}^{1} F(x,\theta _{1}) \,dx}, \frac{\theta _{2}^{2}}{2 \int _{0}^{1} F(x,\theta _{2}) \,dx} \biggr\} \biggr[} $$

and for every nonpositive continuous function $g:\mathbb{R}\rightarrow \mathbb{R}$, there exists $\delta ^{*}>0$, where

$$ \delta ^{*}:= \min \biggl\{ \frac{\theta _{1}^{2} -2\lambda \int _{0}^{1} F(x,\theta _{1}) \,dx }{2 G^{\theta _{1}}}, \frac{\theta _{2}^{2} -4\lambda \int _{0}^{1} F(x,\theta _{2}) \,dx }{4 G^{\theta _{2}}} \biggr\} , $$

such that, for all $\mu \in [0 , \delta ^{*}[$, problem (1.1) admits at least three distinct weak solutions $u_{i}$ for $i\in \{ 1,2,3\}$ such that $0\leq \| u_{i}\|_{\infty }< \theta _{2}$ for every $i\in \{ 1,2,3\}$.

Proof

Without loss of generality, we can assume $f(x,t)\geq 0$ for every $(x,t)\in [0 , 1] \times \mathbb{R}$. Fix λ, μ, and g as in the conclusion and take Φ and Ψ as in the proof of Theorem 3.1. Arguing as in the proof of Theorem 3.1, we observe that the regularity assumptions of Lemma 2.2 on Φ and Ψ are satisfied. Then, our aim is to verify $\mathrm{(b_{1})}$ and $\mathrm{(b_{2})}$.

To this end, put w as given in (3.5), and let $r_{1}= \frac{\theta _{1}^{2}}{2}$ and $r_{2}=\frac{\theta _{2}^{2}}{2}$. It is obvious that $2r_{1} < \Phi (w) < \frac{r_{2}}{2}$. It follows from $\mu < \delta ^{*}$ and $G_{\eta }=0$ that

$$\begin{aligned} \frac{\sup_{\Phi (u)< r_{1}} \Psi (u)}{r_{1}} &= \frac{\sup_{\Phi (u)< r_{1}} (\int _{0}^{1} F(x,u(x)) \,dx+\frac{\mu }{\lambda } G(u(1)) )}{r_{1}} \\ &\leq \frac{2 \int _{0}^{1} F(x,\theta _{1}) \,dx+ \frac{2\mu }{\lambda } G^{\theta _{1}}}{\theta _{1}^{2}} \\ &< \frac{1}{\lambda }< \frac{2}{3} \frac{ \int _{\frac{3}{4}}^{1} F(x,\eta ) \,dx+ \frac{\mu }{\lambda } G_{\eta }}{\frac{32}{27} \pi ^{4} \eta ^{2}} \leq \frac{2}{3} \frac{\Psi (w)}{\Phi (w)}. \end{aligned}$$

Similarly,

$$\begin{aligned} \frac{2 \sup_{\Phi (u)< r_{2}} \Psi (u)}{r_{2}}&\leq \frac{4\int _{0}^{1} F(x,\theta _{2}) \,dx}{\theta _{2}^{2}}+ \frac{4\mu }{\lambda } \frac{G^{\theta _{2}}}{\theta _{2}^{2}} \\ &< \frac{1}{\lambda } < \frac{2}{3} \frac{\int _{\frac{3}{4}}^{1} F(x,\eta ) \,dx+ \frac{\mu }{\lambda } G_{\eta }}{\frac{32}{27} \pi ^{4} \eta ^{2}} \leq \frac{2}{3} \frac{\Psi (w)}{\Phi (w)}. \end{aligned}$$

This implies that $\mathrm{(b_{1})}$ and $\mathrm{(b_{2})}$ of Lemma 2.2 are verified.

Finally, we verify that assumption $\mathrm{(b_{3})}$ of Lemma 2.2 holds. Let $u_{1}$ and $u_{2}$ be two local minima for $\Phi -\lambda \Psi$. Then, $u_{1}$ and $u_{2}$ are critical points for $\Phi -\lambda \Psi$, and so they are weak solutions for problem (1.1). We claim that the weak solutions obtained are nonnegative. Indeed, if $u_{0}$ is a weak solution of problem (1.1), then one has

$$ \int _{0}^{1} u_{0}''(x){v}'' (x) \,dx=\lambda \int _{0}^{1} f\bigl(x,u_{0}(x) \bigr){v}(x) \,dx-\mu g\bigl(u_{0}(1)\bigr){v}(1) $$

for all $v\in X$. Arguing by a contradiction, assume that the set $A:=\{ x\in [0 , 1] : u_{0}(x)< 0\}$ is nonempty and of positive measure. Put ${v_{0}}:=\min \{ 0 , u_{0}\}$. Clearly, $v_{0}\in X$. So, taking into account that $u_{0}$ is a weak solution and by choosing $v=v_{0}$, from our sign assumptions on the data, one has

$$ \int _{A} \bigl(u_{0}''(x) \bigr)^{2}\,dx=\lambda \int _{A} f\bigl(x,u_{0}(x)\bigr){u_{0}}(x) \,dx-\mu g\bigl(u_{0}(1)\bigr){u_{0}}(1)\leq 0. $$

Hence, $u_{0}\equiv 0$ on A which is absurd. Then we deduce $u_{1}(x) \geq 0$ and $u_{2}(x)\geq 0$ for each $x\in [0 , 1]$. Thus, it follows that $su_{1}+(1-s)u_{2}\geq 0$ for all $s\in [0,1]$, and that

$$ \Psi \bigl(su_{1} + (1-s) u_{2}\bigr)= \int _{0}^{1} F\bigl(x, su_{1}(x) + (1-s) u_{2}(x)\bigr) \,dx +\frac{\mu }{\lambda } G\bigl(su_{1}(1) + (1-s) u_{2}(1)\bigr)\geq 0. $$

So, also $\mathrm{(b_{3})}$ holds. From Lemma 2.2, for every

$$ \lambda \in {\biggl]\frac{3}{2}\frac{\Phi (w)}{\Psi (w)}, \min \biggl\{ \frac{r_{1}}{\sup_{\Phi (u)< r_{1}}\Psi (u)}, \frac{\frac{r_{2}}{2}}{\sup_{\Phi (u)< r_{2}}\Psi (u)} \biggr\} \biggr[}, $$

the functional $\Phi -\lambda \Psi$ has at least three distinct critical points which are the solutions of problem (1.1) and the conclusion is achieved. □

4 Conclusion

By using as the main tools two critical point theorems presented recently in the works [3] and [6], we proved two multiplicity properties (Theorems 3.1 and 3.2) that guarantee the existence of an open interval $]\lambda ',\lambda ''[$ and $\tilde{\delta }> 0$ such that, for each $\lambda \in {]\lambda ',\lambda ''[}$ and for each $\mu \in [0,\tilde{\delta }[$, a class of fourth-order boundary value problems depending on parameters λ and μ (problem (1.1)) admits at least three distinct weak solutions.