- Research
- Open Access
Perturbed nonlocal fourth order equations of Kirchhoff type with Navier boundary conditions
- Shapour Heidarkhani1Email author,
- Amjad Salari2,
- Giuseppe Caristi3 and
- David Barilla3
- Received: 5 April 2017
- Accepted: 25 May 2017
- Published: 6 June 2017
Abstract
We investigate the existence of multiple solutions for perturbed nonlocal fourth-order equations of Kirchhoff type under Navier boundary conditions. We give some new criteria for guaranteeing that the perturbed fourth-order equations of Kirchhoff type have at least three weak solutions by using a variational method and some critical point theorems due to Ricceri. We extend and improve some recent results. Finally, by presenting two examples, we ensure the applicability of our results.
Keywords
- three solutions
- Kirchhoff-type equation
- perturbed fourth-order boundary value problem
- variational methods
- critical point theory
MSC
- 35J20
- 35J40
- 35J35
- 31B30
- 35G30
- 74H20
- 35Q35
1 Introduction
In this paper we study the following perturbed nonlocal fourth-order problem of Kirchhoff type under Navier boundary condition:
where \(\Omega \subset \mathbb{R}^{N}\) (\(N\geq 1\)) is a bounded smooth domain,
in which \(p>\max \lbrace 1,\frac{N}{2} \rbrace \), \(\varrho >0\) and \(M:[0, +\infty [\,\to \mathbb{R}\) is a continuous function such that there are two positive constants \(m_{0}\) and \(m_{1}\) with \(m_{0} \leq M(t)\leq m_{1}\) for all \(t\geq 0\), and \(\lambda >0\), \(\mu \geq 0\) and \(f,g:\Omega \times \mathbb{R}\to \mathbb{R}\) are two \(\mathrm{L}^{1}\)-Carathéodory functions.
$$ \left \{ \textstyle\begin{array}{l@{\quad}l} T(u)= \lambda f(x,u)+\mu g(x,u),&\mbox{in }\Omega , \\ u=\Delta u=0,&\mbox{on } \partial \Omega , \end{array}\displaystyle \right . \hspace{30pt} (P^{f,g}_{\lambda ,\mu}) $$
$$ T(u)=\Delta \bigl( \vert \Delta u \vert ^{p-2}\Delta u \bigr) - \biggl[ M \biggl( \int_{\Omega }\vert \nabla u \vert ^{p}\,\mathrm{d}x \biggr) \biggr] ^{p-1} \Delta_{p}u+\varrho \vert u \vert ^{p-2}u, $$
The problem (\(P^{f,g}_{\lambda ,\mu}\)) is related to the stationary problem of a model introduced by Kirchhoff [1]. More precisely, Kirchhoff introduced a model given by the following equation:
where ρ is the mass density, \(\rho_{0}\) is the initial tension, h is the area of the cross section, E is the Young modulus of the material and L is the length of the string, which extends the classical D’Alembert’s wave equation for free vibrations of elastic strings. Kirchhoff’s model takes into account the length changes of the string produced by transverse vibrations. Nonlinear Kirchhoff model can also be used for describing the dynamics of an axially moving string. In recent years, axially moving string-like continua such as wires, belts, chains and band-saws, have been subjects of the study of researchers (see [2]).
$$ \rho \frac{\partial^{2}u}{\partial t^{2}}- \biggl(\frac{\rho_{0}}{h}+ \frac{E}{2L} \int_{0}^{L} \biggl\vert \frac{\partial u}{\partial x} \biggr\vert ^{2}\,\mathrm{d}x \biggr)\frac{\partial^{2}u}{\partial x^{2}}=0, $$
(1.1)
The problem (1.1) was developed into the form
where \(M(s)=as+b\), \(a,b>0\). After that, many authors studied the following nonlocal elliptic boundary value problem:
Problems like (1.3) can be used for modeling several physical and biological systems where u describes a process which depends on the average of itself, such as the population density, see [3]. There are a number of papers concerned with Kirchhoff-type boundary value problem, for instance, see [4–12]. For example, in an interesting paper [12], Ricceri established the existence of at least three weak solutions to a class of Kirchhoff-type doubly eigenvalue boundary value problems. In [5] employing a three critical point theorem due to Ricceri, the authors discussed the existence of at least three weak solutions for Kirchhoff-type problems involving two parameters. In [10] Molica Bisci and Rădulescu, employing mountain pass results, obtained the existence of solutions to nonlocal equations involving the p-Laplacian. More precisely, they proved the existence of at least one nontrivial weak solution and under additional assumptions, the existence of infinitely many weak solutions. In [9], based on an abstract linking theorem for smooth functionals, they also established a multiplicity result on the existence of weak solutions for a nonlocal Neumann problem driven by a nonhomogeneous elliptic differential operator. The existence and multiplicity of stationary problems of Kirchhoff type were also studied in some recent papers via variational methods like the symmetric mountain pass theorem in [13] and via a three critical point theorem in [14]. Moreover, in [15, 16] some evolutionary higher order Kirchhoff problems, mainly focusing on the qualitative properties of solutions, were treated.
$$ u_{tt}-M \biggl( \int_{\Omega }\vert \nabla u \vert ^{2}\,\mathrm{d}x \biggr) \Delta u=f(x,u), $$
(1.2)
$$ -M \biggl( \int_{\Omega }\vert \nabla u \vert ^{2}\,\mathrm{d}x \biggr) \Delta u=f(x,u). $$
(1.3)
We refer to [17–28] for related nonlocal problems concerning the variational analysis of solutions of some classes of boundary value problems.
Fourth-order boundary value problems which describe the deformations of an elastic beam in an equilibrium state whose both ends are simply supported have been extensively studied in the literature. Recently, the existence of solutions to fourth-orderboundary value problems have been studied in many papers, and we refer the reader to the papers [29–37] and the references therein. For example, Liu et al. in [35], employing variational methods, studied the existence and multiplicity of nontrivial solutions for fourth-order elliptic equations. In [30, 32], based on variational methods and critical point theory, the existence of multiple solutions for \((p_{1},\ldots ,p_{n})\)-biharmonic systems was discussed. Molica Bisci and Repovs̆ in [37], exploiting variational methods, investigated the existence of multiple weak solutions for a class of elliptic Navier boundary problems involving the p-biharmonic operator and presented a concrete example of an application.
The problem (\(P^{f,g}_{\lambda ,\mu}\)) models the bending equilibrium of simply supported extensible beams on nonlinear foundations. The function f represents the force that the foundation exerts on the beam and \(M ( \int_{\Omega }\vert \nabla u \vert ^{p}\,\mathrm{d}x ) \) models the effects of the small changes in the length of the beam. Recently, many researchers have paid their attention to fourth-order Kirchhoff-type problems; we refer the reader to [38–42] and the references therein. In [41], using the mountain pass theorem, Wang and An established the existence and multiplicity of solutions for the following fourth-order nonlocal elliptic problem:
In particular, in [38] using variational methods and critical point theory, multiplicity results of nontrivial and nonnegative solutions for the parametric version of the problem (\(P^{f,g}_{\lambda ,\mu}\)) was established. In [39], employing two three critical point theorems, the existence of three distinct weak solutions for the problem (\(P^{f,g}_{\lambda ,\mu}\)) was ensured.
$$ \left \{ \textstyle\begin{array}{l@{\quad}l} \Delta^{2} u-M (\int_{\Omega }\vert \nabla u \vert ^{2}\,\mathrm{d}x )\Delta u=\lambda f(x,u)&\mbox{in }\Omega, \\ u=\Delta u=0&\mbox{on }\partial \Omega . \end{array}\displaystyle \right . $$
Inspired by the above facts, in the present paper we are interested in looking for the existence of at least three weak solutions for the problem (\(P^{f,g}_{\lambda ,\mu}\)) for appropriate values of the parameters λ and μ belonging to real intervals. Employing variational methods and two three critical point theorems due to Ricceri [43, 44], we establish two existence results for the problem (\(P^{f,g}_{\lambda ,\mu}\)). Two examples are presented to illustrate our main results.
For more details on the subject, we also refer the reader to [45, 46].
2 Preliminaries
Our main tools are two three critical point theorems obtained by Ricceri [43, 44]. They are as follows.
If X is a real Banach space denoted by \(\mathcal{W}_{X}\), the class of all functionals \(\Phi :X\rightarrow \mathbb{R}\) possesses the following property:
If \(\{u_{n}\}\) is a sequence in X converging weakly to \(u\in X\) and \(\liminf_{n\rightarrow \infty } \Phi (u_{n}) \leq \Phi (u)\), then \(\{u_{n}\}\) has a subsequence converging strongly to u.
Remark 2.1
If X is uniformly convex and \(g:[0,+\infty )\rightarrow \mathbb{R}\) is a continuous and strictly increasing function, then, by a classical result, the functional \(u\rightarrow g(\Vert u \Vert )\) belongs to the class \(\mathcal{W}_{X}\).
Theorem 2.1
[43]
Let
X
be a separable and reflexive real Banach space; let
\(\Phi :X\rightarrow \mathbb{R}\)
be a coercive, sequentially weakly lower semicontinuous
\(\mathrm{C}^{1}\)-functional, belonging to
\(\mathcal{W} _{X}\), bounded on each bounded subset of
X
and whose derivative admits a continuous inverse on
\(X^{*}\); \(J:X\rightarrow \mathbb{R}\)
be a
\(\mathrm{C}^{1}\)-functional with compact derivative. Assume that Φ has a strict local minimum
\(u_{0}\)
with
\(\Phi (u_{0})=J(u_{0})=0\). Finally, setting
assume that
\(\rho <\sigma \). Then, for each compact interval
\([c,d]\subset (\frac{1}{\sigma },\frac{1}{\rho })\) (with the conventions
\(\frac{1}{0} =\infty , \frac{1}{\infty }=0\)), there exists
\(R>0\)
with the following property: for every
\(\lambda \in [c,d]\)
and every
\(\mathrm{C}^{1}\)-functional
\(\Psi :X\rightarrow \mathbb{R}\)
with compact derivative, there exists
\(\gamma >0\)
such that, for each
\(\mu \in [0, \gamma ]\),
has at least three solutions in
X
whose norms are less than R.
$$\begin{aligned}& \rho =\max \biggl\{ 0,\limsup_{\Vert u \Vert \rightarrow +\infty }\frac{J(u)}{ \Phi (u)}, \limsup _{u \rightarrow u_{0}}\frac{J(u)}{\Phi (u)} \biggr\} , \\& \sigma =\sup_{u\in \Phi^{-1}(0,+\infty )}\frac{J(u)}{\Phi (u)}, \end{aligned}$$
$$ \Phi '(u)=\lambda J'(u)+\mu \Psi '(u) $$
Theorem 2.2
[44]
Let
X
be a reflexive real Banach space and
\(I\subseteq \mathbb{R}\)
be an interval; let
\(\Phi :X\to \mathbb{R}\)
be a sequentially weakly lower semi-continuous
\(\mathrm{C}^{1}\)-functional, bounded on each bounded subset of
X, whose derivative admits a continuous inverse on
\(X^{*}\); let
\(J:X\to \mathbb{R}\)
be a
\(\mathrm{C}^{1}\)-functional with compact derivative. Assume that
for all
\(\lambda \in I\), and that there exists
\(\rho \in \mathbb{R}\)
such that
Then there exist a nonempty open set
\(A\subseteq I\)
and a positive number
\(R'\)
with the following property: for every
\(\lambda \in A\)
and every
\(\mathrm{C}^{1}\)
functional
\(\Psi :X\to \mathbb{R}\)
with compact derivative, there exists
\(\delta >0\)
such that, for each
\(\mu \in [0, \delta ]\), the equation
\(\Phi '(u)-\lambda J'(u)-\mu \Psi '(u)=0\)
has at least three solutions in
X
whose norms are less than
\(R'\).
$$ \lim_{\Vert u \Vert \to +\infty } \bigl(\Phi (u)-\lambda J(u) \bigr)=+\infty $$
$$ \sup_{\lambda \in I}\inf_{u\in X} \bigl(\Phi (u)+\lambda \bigl(\rho -J(u) \bigr) \bigr)< \inf_{u\in X}\sup _{\lambda \in I} \bigl(\Phi (u)+\lambda \bigl(\rho -J(u) \bigr) \bigr). $$
Proposition 2.3
[47]
Let
X
be a nonempty set and Φ, J
be two real functions on
X. Assume that there are
\(r>0\)
and
\(u_{0},u_{1}\in X\)
such that
Then, for each
ρ
satisfying
one has
$$ \Phi (u_{0})=J(u_{0})=0,\quad \quad \Phi (u_{1})>r,\quad \quad \sup_{u\in \Phi^{-1}(-\infty ,r]}J(u)< r \frac{J(u_{1})}{\Phi (u_{1})} . $$
$$ \sup_{u\in \Phi^{-1}(-\infty ,r]}J(u)< \rho < r\frac{J(u_{1})}{\Phi (u _{1})}, $$
$$ \sup_{\lambda \geq 0}\inf_{u\in X} \bigl(\Phi (u)+\lambda \bigl(\rho -J(u) \bigr) \bigr)< \inf_{u\in X}\sup _{\lambda \geq 0} \bigl(\Phi (u)+\lambda \bigl(\rho -J(u) \bigr) \bigr). $$
We refer the reader to the paper [48, 49] in which Theorems 2.1 and 2.2 were successfully employed to ensure the existence of at least three solutions for perturbed second-order Hamiltonian systems with impulsive effects. We also refer the readers to [50] in which Theorems 2.1 and 2.2 were successfully employed to ensure the existence of three solutions for perturbed Kirchhoff-type p-Laplacian discrete problems, and we refer the readers to [51] in which Theorem 2.1 was successfully employed to ensure the existence of three solutions for impulsive perturbed elastic beam fourth-order equations of Kirchhoff type.
Here and in the sequel, \(\rm E\) will denote the space \(W^{2,p}(\Omega )\cap W_{0}^{1,p}(\Omega )\) endowed with the norm
Put
For \(p>\max \{1,\frac{N}{2}\}\), since the embedding \(\rm E\hookrightarrow C^{0}(\overline{\Omega })\) is compact, one has \(k<+\infty \).
$$ \Vert u \Vert := \biggl( \int_{\Omega } \bigl( \bigl\vert \Delta u(x) \bigr\vert ^{p}+ \bigl\vert \nabla u(x) \bigr\vert ^{p} + \bigl\vert u(x) \bigr\vert ^{p} \bigr)\,\mathrm{d}x \biggr)^{\frac{1}{p}}. $$
$$ k=\sup_{u\in X\setminus \{0\}}\frac{\max_{x\in \overline{\Omega }}\vert u(x) \vert }{ \Vert u \Vert }. $$
(2.1)
Let \(f,g:\Omega \times \mathbb{R}\to \mathbb{R}\) be two \(\mathrm{L}^{1}\)-Carathéodory functions and \(M:[0, +\infty [\,\to \mathbb{R}\) be a continuous function such that there are two positive constants \(m_{0}\) and \(m_{1}\) with \(m_{0}\leq M(t)\leq m_{1}\) for all \(t\geq 0\).
Put
and
Set
and
$$\begin{aligned}& F(x,t):= \int_{0}^{t} f(x,\xi )\,\mathrm{d}\xi \quad \mbox{for all } (x,t) \in \Omega \times \mathbb{R}, \\& G(x,t):= \int_{0}^{t} g(x,\xi )\,\mathrm{d}\xi \quad \mbox{for all } (x,t) \in \Omega \times \mathbb{R} \end{aligned}$$
$$ \widehat{M}(t):= \int_{0}^{t} \bigl[ M(s) \bigr] ^{p-1}\,{\mathrm{d}}s \quad \mbox{for all } t\geq 0. $$
$$ M^{-} := \min \bigl\lbrace 1,m_{0}^{p-1} ,\varrho \bigr\rbrace $$
$$ M^{+}:=\max \bigl\lbrace 1,m_{1}^{p-1} ,\varrho \bigr\rbrace . $$
We say that a function \(u\in {\mathrm{E}}\) is a (weak) solution of the problem (\(P^{f,g}_{\lambda ,\mu}\)) if
for every \(v\in {\mathrm{E}}\).
$$\begin{aligned}& \int_{\Omega }\vert \Delta u \vert ^{p-2}\Delta u\Delta v\,\mathrm{d}x + \biggl[ M \biggl( \int_{\Omega }\vert \nabla u \vert ^{p} \,{\mathrm{d}}x \biggr) \biggr] ^{p-1} \int_{\Omega }\vert \nabla u \vert ^{p-2}\nabla u\nabla v\,\mathrm{d}x \\& \quad {}+\varrho \int_{\Omega }\vert u \vert ^{p-2}uv\,\mathrm{d}x - \lambda \int_{\Omega }f(x,u)v\,\mathrm{d}x-\mu \int_{\Omega }g(x,u)v\,\mathrm{d}x=0 \end{aligned}$$
Now, for every \(u\in {\mathrm{E}}\), we define
and
$$\begin{aligned}& \Phi (u)=\frac{1}{p} \int_{\Omega } \bigl\vert \Delta u(x) \bigr\vert ^{p}{ \mathrm{d}}x+ \frac{1}{p}\widehat{M} \biggl[\int_{\Omega } \bigl\vert \nabla u(x) \bigr\vert ^{p} { \mathrm{d}}x \biggr] +\frac{\varrho }{p}\int_{\Omega } \bigl\vert u(x) \bigr\vert ^{p}\,{\mathrm{d}}x, \end{aligned}$$
(2.2)
$$\begin{aligned}& J(u)= \int_{\Omega }F \bigl(x,u(x) \bigr)\,\mathrm{d}x \end{aligned}$$
(2.3)
$$ \Psi (u)= \int_{\Omega }G \bigl(x,u(x) \bigr)\,\mathrm{d}x. $$
(2.4)
Standard arguments show that \(I=:\Phi -\mu \Psi -\lambda J\) is a Gâteaux differentiable functional whose Gâteaux derivative at the point \(u\in {\mathrm{E}}\) is given by
for all \(u,v\in {\mathrm{E}}\). We observe that a vector \(u\in {\mathrm{E}}\) is a solution of the problem (\(P^{f,g}_{\lambda ,\mu}\)) if and only if u is a critical point of the function I.
$$\begin{aligned} I'(u) (v)&= \int_{\Omega }\vert \Delta u \vert ^{p-2}\Delta u\Delta v \,\mathrm{d}x + \biggl[ M \biggl( \int_{\Omega }\vert \nabla u \vert ^{p} \,{\mathrm{d}}x \biggr) \biggr] ^{p-1} \int_{\Omega }\vert \nabla u \vert ^{p-2}\nabla u\nabla v\,\mathrm{d}x \\ &\quad {}+\varrho \int_{\Omega }\vert u \vert ^{p-2}uv\,\mathrm{d}x - \lambda \int_{\Omega }f(x,u)v\,\mathrm{d}x-\mu \int_{\Omega }g(x,u)v\,\mathrm{d}x \end{aligned}$$
3 Main results
In this section, we formulate our main results.
First we give the following application of Theorem 2.1 as our first main result. Let
and \(\lambda_{2}= \frac{1}{\max \{ 0,\lambda_{0},\lambda_{\infty } \} }\), where
and
$$\begin{aligned} \lambda_{1}&=\inf \biggl\{ \frac{\int_{\Omega }\vert \Delta u(x) \vert ^{p} \,{\mathrm{d}}x+ \widehat{M} [ \int_{\Omega }\vert \nabla u(x) \vert ^{p} \,{\mathrm{d}}x ] +\varrho \int_{\Omega }\vert u(x) \vert ^{p}\,{\mathrm{d}}x}{p \int_{\Omega }F(x,u(x))\,\mathrm{d}x}\,\colon \\ & \quad {}u\in {\mathrm{E}}, \int_{\Omega }F \bigl(x,u(x) \bigr)\,\mathrm{d}x>0 \biggr\} \end{aligned}$$
$$\begin{aligned} \lambda_{0}=\limsup_{\vert u \vert \rightarrow 0} \frac{p\int_{\Omega }F(x,u(x)) \,\mathrm{d}x}{\int_{\Omega }\vert \Delta u(x) \vert ^{p}\,{\mathrm{d}}x+ \widehat{M} [ \int_{\Omega }\vert \nabla u(x) \vert ^{p} \,{\mathrm{d}}x ] + \varrho \int_{\Omega }\vert u(x) \vert ^{p}\,{\mathrm{d}}x} \end{aligned}$$
$$\begin{aligned} \lambda_{\infty }=\limsup_{\Vert u \Vert \rightarrow + \infty }\frac{p \int_{\Omega }F(x,u(x))\,\mathrm{d}x}{\int_{\Omega }\vert \Delta u(x) \vert ^{p}\,{\mathrm{d}}x+ \widehat{M} [ \int_{\Omega }\vert \nabla u(x) \vert ^{p} \,{\mathrm{d}}x ] +\varrho \int_{\Omega }\vert u(x) \vert ^{p}\,{\mathrm{d}}x}. \end{aligned}$$
Theorem 3.1
Assume that
Then, for each compact interval
\([c,d]\subset (\lambda_{1},\lambda _{2})\), there exists
\(R> 0\)
with the following property: for every
\(\lambda \in [c,d]\)
and every
\(\mathrm{L}^{1}\)-Carathéodory function
\(g:\Omega \times \mathbb{R}\to \mathbb{R}\), there exists
\(\gamma >0\)
such that, for each
\(\mu \in [0,\gamma ]\), the problem (\(P^{f,g}_{\lambda ,\mu}\)) has at least three weak solutions whose norms in E are less than
R.
- \((\mathcal{A}_{1})\) :
-
there exists a constant \(\varepsilon >0\) such that$$ \max \biggl\{ \limsup_{u\rightarrow 0}\frac{\max_{x\in \overline{\Omega }}F(x,u(x))}{\vert u \vert ^{p}} ,\,\limsup _{\vert u \vert \to +\infty } \frac{ \max_{x\in \overline{\Omega }}F(x,u(x))}{\vert u \vert ^{p}} \biggr\} < \varepsilon ; $$
- \((\mathcal{A}_{2})\) :
-
there exists a function \(w\in {\mathrm{E}}\) such thatand$$ K_{w}:= \int_{\Omega } \bigl\vert \Delta w(x) \bigr\vert ^{p}\,{\mathrm{d}}x+\widehat{M} \biggl[ \int_{\Omega } \bigl\vert \nabla w(x) \bigr\vert ^{p}\,{ \mathrm{d}}x \biggr] + \varrho \int_{\Omega } \bigl\vert w(x) \bigr\vert ^{p}\,{\mathrm{d}}x\neq 0 $$$$ \varepsilon < \frac{M^{-}\int_{\Omega }F(x,w(x))\,\mathrm{d}x}{k^{p}K_{w}}. $$
Proof
Take \(X=\mathrm{E}\). Clearly, E is a separable and uniformly convex Banach space. Let the functionals Φ, J and Ψ be as given in (2.2), (2.3) and (2.4), respectively. The functional Φ is \(\mathrm{C}^{1}\), and by [38], Proposition 2.3, its derivative admits a continuous inverse on \(X^{*}\). Moreover, since \(m_{0}\leq K(s)\leq m_{1}\) for all \(s\in [0,+\infty[\), from (2.2) we have
for all \(u \in X\), it follows \(\lim_{\Vert u \Vert \to +\infty }\Phi (u)=+ \infty \), namely, Φ is coercive. Furthermore, Φ is sequentially weakly lower semicontinuous. Moreover, let A be a bounded subset of X. That is, there exists a constant \(c>0\) such that \(\Vert u \Vert \leq c\) for each \(u\in A\). Then, by (3.1), we have
Hence Φ is bounded on each bounded subset of X. Furthermore, by Remark 2.1, \(\Phi \in \mathcal{W}_{X}\). The functionals J and Ψ are two \(\mathrm{C}^{1}\)-functionals with compact derivatives. Moreover, Φ has a strict local minimum 0 with \(\Phi (0)=J(0)=0\). In view of \((\mathcal{A}_{1})\), there exist \(\tau_{1}\), \(\tau_{2}\) with \(0<\tau_{1}<\tau_{2}\) such that
for every \(x\in \Omega \) and every u with \(\vert u \vert \in [0, \tau_{1}) \cup (\tau_{2}, +\infty )\). Since \(F(x,u)\) is continuous on \(\Omega \times \mathbb{R}\), it is bounded on \(x\in \Omega \) and \(\vert u \vert \in [\tau_{1},\tau_{2}]\). Thus we can choose \(\eta >0\) and \(\upsilon >p\) such that
for all \((k,u)\in \Omega \times \mathbb{R}\). So, by (2.1), we have
for all \(u\in X\). Hence, from (3.3) we have
Moreover, by using (3.2), for each \(u\in {\mathrm{E}}\setminus \{0 \}\), we obtain
So, we get
In view of (3.4) and (3.5), we have
Assumption \((\mathcal{A}_{2})\) in conjunction with (3.6) yields
Thus, all the hypotheses of Theorem 2.1 are satisfied. Clearly, \(\lambda_{1}=\frac{1}{\sigma }\) and \(\lambda_{2}=\frac{1}{ \rho }\). Then, using Theorem 2.1, for each compact interval \([c,d]\subset (\lambda_{1},\lambda_{2})\), there exists \(R> 0\) with the following property: for every \(\lambda \in [c,d]\) and every \(\mathrm{L}^{1}\)-Carathéodory function \(g:\Omega \times \mathbb{R} \to \mathbb{R}\), there exists \(\gamma >0\) such that, for each \(\mu \in [0,\gamma ]\), the problem (\(P^{f,g}_{\lambda ,\mu}\)) has at least three solutions whose norms in X are less than R. □
$$\begin{aligned} \frac{M^{-}}{p}\Vert u \Vert ^{p}\leq \Phi (u) \leq \frac{M^{+}}{p}\Vert u \Vert ^{p} \end{aligned}$$
(3.1)
$$ \bigl\vert \Phi (u) \bigr\vert \leq \frac{M^{+}}{p}c^{p}. $$
$$ F(x,u)\leq \varepsilon \vert u \vert ^{p} $$
(3.2)
$$ F(x,u)\leq \varepsilon \vert u \vert ^{p}+\eta \vert u \vert ^{\upsilon } $$
$$ J(u)\leq \varepsilon k^{p}\Vert u \Vert ^{p}+ \eta k^{\upsilon }\Vert u \Vert ^{\upsilon } $$
(3.3)
$$ \limsup_{\vert u \vert \rightarrow 0}\frac{J(u)}{\Phi (u)}\leq \frac{pk^{p} \varepsilon }{M^{-}}. $$
(3.4)
$$\begin{aligned} \frac{J(u)}{\Phi (u)} &=\frac{\int_{\vert u \vert \leq \tau_{2}}F(x,u)\,\mathrm{d}x}{ \Phi (u)} +\frac{\int_{\vert u \vert >\tau_{2}}F(x,u)\,\mathrm{d}x}{\Phi (u)} \\ &\leq \frac{\sup_{x\in \Omega ,\vert u \vert \in [0,\tau_{2}]}F(x,u)}{\Phi (u)} +\frac{\varepsilon \Vert u \Vert ^{p}}{k^{p}\Phi (u)}\leq \frac{p\sup_{x\in \Omega ,\vert u \vert \in [0,\tau_{2}]}F(x,u)}{\Vert u \Vert ^{p}}+ \frac{pk^{p}\varepsilon }{M^{-}}. \end{aligned}$$
$$ \limsup_{\Vert u \Vert \rightarrow + \infty }\frac{J(u)}{\Phi (u)}\leq \frac{pk ^{p}\varepsilon }{M^{-}}. $$
(3.5)
$$ \rho =\max \biggl\{ 0,\,\limsup_{\Vert u \Vert \rightarrow + \infty } \frac{J(u)}{ \Phi (u)},\, \limsup_{u\rightarrow 0}\frac{J(u)}{\Phi (u)} \biggr\} \leq \frac{pk^{p}\varepsilon }{M^{-}}. $$
(3.6)
$$\begin{aligned} \sigma &=\sup_{u\in \Phi^{-1}(0,+\infty )}\frac{J(u)}{\Phi (u)} \\ &= \sup_{X\setminus \{0\}}\frac{J(u)}{\Phi (u)}\geq \frac{\int_{\Omega }F(x,w(x))}{ \Phi (w(x))} \\ &= \frac{p\int_{\Omega }F(x,w(x))\,\mathrm{d}x}{K_{w}}>\frac{pk^{p}\varepsilon }{M^{-}}\geq \rho . \end{aligned}$$
Another announced application of Theorem 2.1 reads as follows.
Theorem 3.2
Assume that
and
Then, for each compact interval
\([c,d]\subset (\lambda_{1},+\infty )\), there exists
\(R>0\)
with the following property: for every
\(\lambda \in [c,d]\)
and every
\(\mathrm{L}^{1}\)-Carathéodory function
\(g:\Omega \times \mathbb{R}\to \mathbb{R}\), there exists
\(\gamma >0\)
such that for each
\(\mu \in [0,\gamma ]\), the problem (\(P^{f,g}_{\lambda ,\mu}\)) has at least three weak solutions whose norms in E are less than
R.
$$ \max \biggl\{ \limsup_{u\rightarrow 0}\frac{\max_{x\in \overline{\Omega }}F(x,u(x))}{\vert u \vert ^{p}}, \,\limsup_{\vert u \vert \to +\infty } \frac{ \max_{x\in \overline{\Omega }}F(x,u(x))}{\vert u \vert ^{p}} \biggr\} \leq 0 $$
(3.7)
$$ \sup_{u\in {\mathrm{E}}}\frac{M^{-}\int_{\Omega }F(x,u(x))\,\mathrm{d}x}{k^{p}( \int_{\Omega }\vert \Delta u(x) \vert ^{p}\,{\mathrm{d}}x+ \widehat{M} [ \int _{\Omega }\vert \nabla u(x) \vert ^{p}\,{\mathrm{d}}x ] +\varrho \int_{\Omega }\vert u(x) \vert ^{p}\,{\mathrm{d}}x)}>0. $$
(3.8)
Proof
In view of (3.7), there exist an arbitrary \(\varepsilon >0\) and \(\tau_{1}\), \(\tau_{2}\) with \(0<\tau_{1}<\tau_{2}\) such that
for every \(x\in \Omega \) and every u with \(\vert u \vert \in [0, \tau_{1}) \cup (\tau_{2}, +\infty )\). Since \(F(x,u)\) is continuous on \(\Omega \times \mathbb{R}\), it is bounded on \(x\in \Omega \) and \(\vert u \vert \in [\tau_{1},\tau_{2}]\). Thus we can choose \(\eta >0\) and \(\upsilon >p\) in a manner that
for all \((x,u)\in \Omega \times \mathbb{R}\). So, by the same process as that in the proof of Theorem 3.1, we have Relations (3.4) and (3.5). Since ε is arbitrary, (3.4) and (3.5) give
Then, with the notation of Theorem 2.1, we have \(\rho =0\). By (3.8), we also have \(\sigma >0\). In this case, clearly \(\lambda_{1}=\frac{1}{\sigma }\) and \(\lambda_{2}=+\infty \). Thus, by using Theorem 2.1, the result is achieved. □
$$ F(x,u)\leq \varepsilon \vert u \vert ^{p} $$
$$ F(x,u)\leq \varepsilon \vert u \vert ^{p}+\eta \vert u \vert ^{\upsilon } $$
$$ \max \biggl\{ 0,\,\limsup_{\Vert u \Vert \rightarrow + \infty }\frac{J(u)}{ \Phi (u)},\, \limsup _{u\rightarrow 0}\frac{J(u)}{\Phi (u)} \biggr\} \leq 0. $$
Now we formulate the following applications of Theorem 2.2 as our second main result.
Theorem 3.3
Assume that
Then there exist a nonempty open set
\(A\subset [0,+\infty )\)
and a positive number
\(R'\)
with the following property: for every
\(\lambda \in A\)
and every continuous function
\(g:\Omega \times \mathbb{R}\to \mathbb{R}\), there exists
\(\delta >0\)
such that, for each
\(\mu \in [0,\delta ]\), the problem (\(P^{f,g}_{\lambda ,\mu}\)) has at least three weak solutions whose norms in E are less than
\(R'\).
- \((\mathcal{B}_{1})\) :
-
there exist two positive functions \(\nu ,\xi \in {\mathrm{L}}^{1}(\Omega ,\mathbb{R})\) and \(\alpha \in [0,p)\) such that$$ \bigl\vert F(x,u) \bigr\vert \leq \nu (x)\vert u \vert ^{\alpha }+ \xi (x)\quad \textit{for all } u \in \mathbb{R} \textit{ and } x\in \Omega ; $$
- \((\mathcal{B}_{2})\) :
-
there exist a positive constant r and \(w\in {\mathrm{E}}\) such that \(K_{w}>r\), where \(K_{w}\) is as given in Assumption \((\mathcal{A}_{2})\) in Theorem 3.1, and$$ \max_{x\in \overline{\Omega },\vert u \vert \leq \sqrt[p]{\frac{pr}{M^{-}}}}F(x,u)< \frac{r \int_{\Omega }F(x,w(x))\,\mathrm{d}x}{\operatorname{meas}(\Omega )K_{w}}. $$
Proof
Take \(X=\mathrm{E}\). Let the functionals Φ and J be as given in (2.2) and (2.3), respectively. For any \(\lambda \geq 0\) and \(u\in {\mathrm{E}}\), by \((\mathcal{B}_{1})\), we have
Since \(\alpha < p\), one has \(\lim_{\Vert u \Vert \to +\infty }(\Phi (u)-\lambda J(u))=+\infty \) for all \(\lambda \geq 0\). If \(\Phi (u)\leq r\), we have \(\Vert u \Vert \leq \sqrt[p]{\frac{pr}{M^{-}}}\), that is,
Therefore,
It is clear that \(\Phi (0)=J(0)=0\) and owing to \((\mathcal{B}_{2})\) and (3.9), \(\Phi (w)>r\) and
Thus we can fix ρ such that
Now, from Proposition 2.3, we obtain
Therefore, by Theorem 2.2, for each compact interval \([a,b]\subseteq (\lambda_{1},\lambda_{2})\), there exists \(R'>0\) with the following property: for every \(\lambda \in [a,b]\) and every \(\mathrm{L}^{1}\)-Carathéodory function \(g:\Omega \times \mathbb{R} \to \mathbb{R}\), there exists \(\delta >0\) such that, for each \(\mu \in [0,\delta ]\), \(\Phi '(u)-\lambda J'(u)-\mu \Psi '(u)=0\) has at least three solutions in E. Hence, the problem (\(P^{f,g}_{\lambda ,\mu}\)) has at least three weak solutions whose norms are less than \(R'\). □
$$\begin{aligned} \Phi (u)-\lambda J(u) &=\frac{1}{p} \int_{\Omega } \bigl\vert \Delta u(x) \bigr\vert ^{p}\,{\mathrm{d}}x+ \frac{1}{p}\widehat{M} \biggl[ \int_{\Omega } \bigl\vert \nabla u(x) \bigr\vert ^{p}\,{ \mathrm{d}}x \biggr] \\ &\quad {}+\frac{\varrho }{p} \int_{\Omega } \bigl\vert u(x) \bigr\vert ^{p}\,{\mathrm{d}}x-\lambda \int_{\Omega }F \bigl(x,u(x) \bigr)\,\mathrm{d}x \\ & \geq \frac{M^{-}}{p}\Vert u \Vert ^{p} -\lambda \int_{\Omega } \bigl( \nu (x) \bigl\vert u(x) \bigr\vert ^{ \alpha }+\xi (x) \bigr) \,\mathrm{d}x \\ &\geq \frac{M^{-}}{p}\Vert u \Vert ^{p} -\lambda k^{\alpha }\Vert u \Vert ^{\alpha }\Vert \nu \Vert _{\mathrm{L}^{1}}-\lambda \Vert \xi \Vert _{\mathrm{L}^{1}}. \end{aligned}$$
$$ \Phi^{-1}(-\infty ,r]\subseteq \biggl\{ u\in X:\max _{x\in \overline{\Omega }} \bigl\vert u(x) \bigr\vert \leq \sqrt[p]{ \frac{pr}{M^{-}}} \biggr\} . $$
$$\begin{aligned} \sup_{u\in \Phi^{-1}(-\infty ,r]}J(u) &\leq \max_{\vert u \vert \leq \sqrt[p]{\frac{pr}{M^{-}}}}J(u) \\ &=\max_{\vert u \vert \leq \sqrt[p]{\frac{pr}{M^{-}}}} \int_{\Omega }F \bigl(x,u(x) \bigr) \,\mathrm{d}x \\ & \leq {\operatorname{meas}}(\Omega ) \max_{x\in \overline{\Omega },\vert u \vert \leq \sqrt[p]{\frac{pr}{M^{-}}}}F(x,u). \end{aligned}$$
(3.9)
$$ \sup_{u\in \Phi^{-1}(-\infty ,r]}J(u)< r\frac{J(w)}{\Phi (w)}. $$
$$ \sup_{u\in \Phi^{-1}(-\infty ,r]}J(u)< \rho < r\frac{J(w)}{\Phi (w)}. $$
$$ \sup_{\lambda \geq 0}\inf_{u\in {\mathrm{E}}} \bigl(\Phi (u)+\lambda \bigl(\rho -J(u) \bigr) \bigr)< \inf_{u\in {\mathrm{E}}}\sup _{\lambda \geq 0} \bigl(\Phi (u)+\lambda \bigl(\rho -J(u) \bigr) \bigr). $$
Now fix \(x^{0}\in \Omega \) and pick \(s>0\) such that \(B(x^{0},s)\subset \Omega \), where \(B(x^{0},s)\) denotes the ball with center at \(x^{0}\) and radius of s. Put
where Γ denotes the gamma function, and
The next two theorems provide sufficient conditions for applying Theorems 3.1 and 3.3, which does not require to know a test function w satisfying \((\mathcal{A}_{2})\) and \((\mathcal{B} _{2})\), respectively.
$$\begin{aligned}& \vartheta_{1}:= \frac{2\pi^{\frac{N}{2}}}{\Gamma (\frac{N}{2})} \int_{\frac{s}{2}}^{s} \biggl\vert \frac{12(N+1)}{s^{3}}r- \frac{24 N}{s^{2}}+\frac{9(N-1)}{s}\frac{1}{r} \biggr\vert ^{p} r^{N-1}\,{\mathrm{d}}r , \\& \vartheta_{2}:= \int_{ B(x^{0},s)\setminus B(x^{0},\frac{s}{2})} \Biggl[ \sum_{i=1}^{N} \biggl( \frac{12(x_{i}-x^{0}_{i})}{s^{3}}-\frac{24(x_{i}-x ^{0}_{i})}{s^{2}}+\frac{9(x_{i}-x^{0}_{i})}{s\ell } \biggr)^{2} \Biggr]^{\frac{p}{2}}\,{\mathrm{d}}x, \\& \vartheta_{3}:=\frac{2\pi^{\frac{N}{2}}}{\Gamma (\frac{N}{2})} \biggl[ \frac{( \frac{s}{2})^{N}}{N} + \int_{\frac{s}{2}}^{s} \biggl\vert \frac{4}{s^{3}}r^{3} -\frac{12}{s^{2}} r^{2} +\frac{9}{s}r -1 \biggr\vert ^{p} r^{N-1}\,{\mathrm{d}}r \biggr] , \end{aligned}$$
$$ L:=\vartheta_{1}+\vartheta_{2}+ \vartheta_{3}. $$
(3.10)
Theorem 3.4
Assume that Assumption
\((\mathcal{A}_{1})\)
in Theorem
3.1
holds and there exists a positive constant
d
such that
Then, for each compact interval
\([c,d]\subset (\lambda_{1},\lambda _{2})\), there exists
\(R> 0\)
with the following property: for every
\(\lambda \in [c,d]\)
and every
\(\mathrm{L}^{1}\)-Carathéodory function
\(g:\Omega \times \mathbb{R}\to \mathbb{R}\), there exists
\(\gamma >0\)
such that, for each
\(\mu \in [0,\gamma ]\), the problem (\(P^{f,g}_{\lambda ,\mu}\)) has at least three weak solutions whose norms in E are less than
R.
- \((\mathcal{A}_{3})\) :
-
\(F(x,t)\geq 0\) for each \(x\in B(x^{0},s) \setminus B(x^{0},\frac{s}{2})\), \(t\in [0,d]\);
- \((\mathcal{A}_{4})\) :
-
\(\vartheta_{1}d^{p} +\widehat{M}(\vartheta _{2}d^{p}) +\varrho \vartheta_{3}d^{p}\neq 0\) and \(\varepsilon <\frac{pM ^{-}\int_{B(x^{0},\frac{s}{2})}F(x,d)\,\mathrm{d}x}{k^{p}(\vartheta_{1}d ^{p}+\widehat{M}(\vartheta_{2}d^{p})+\varrho \vartheta_{3}d^{p})}\).
Proof
We claim that all the assumptions of Theorem 3.1 are fulfilled by choosing w as follows:
where \(\ell =\operatorname{dist}(x,x^{0})=\sqrt{\sum_{i=1}^{N} ( x_{i}-x_{i}^{0} ) ^{2}}\) (see [31, 36]). We have
and
and so that
In particular, since
and
It is easy to see that \(w\in {\mathrm{E}}\), and one has
From Assumptions \((\mathcal{A}_{3})\) and \((\mathcal{A}_{4})\) we observe that Assumption \((\mathcal{A}_{2})\) in Theorem 3.1 is satisfied. Hence, Theorem 3.1 follows the result. □
$$ w(x):=\left \{\textstyle\begin{array}{l@{\quad}l} 0 &\mbox{if } x \in \overline{\Omega }\setminus B(x^{0} , s), \\ d ( \frac{4}{s^{3}}\ell^{3} -\frac{12}{s^{2}}\ell^{2} + \frac{9}{s}\ell -1 ) &\mbox{if } x\in B(x^{0},s)\setminus B(x ^{0},\frac{s}{2}), \\ d &\mbox{if } x \in B(x^{0},\frac{ s}{2}) \end{array}\displaystyle \right . $$
(3.11)
$$\begin{aligned}& \frac{\partial w(x)}{\partial x_{i}} = \textstyle\begin{cases} 0&\mbox{if } x \in \overline{\Omega }\setminus B(x^{0},s)\cup B(x^{0},\frac{s}{2}), \\ d ( \frac{12 \ell (x_{i}-x_{i}^{0})}{s^{3}}-\frac{24(x_{i}-x_{i} ^{0})}{s^{2}}+\frac{9}{s}\frac{(x_{i}-x_{i}^{0})}{\ell } ) & \mbox{if } x\in B(x^{0},s)\setminus B(x^{0},\frac{s}{2}) \end{cases}\displaystyle \end{aligned}$$
$$\begin{aligned}& \frac{\partial^{2} w(x)}{\partial x_{i}^{2}}= \textstyle\begin{cases} 0&\mbox{if } x \in \overline{\Omega }\setminus B(x^{0},s)\cup B(x^{0},\frac{s}{2}), \\ d ( \frac{12}{s^{3}}\frac{(x_{i}-x_{i}^{0})^{2}+\ell^{2}}{\ell }-\frac{24}{s ^{2}}+\frac{9}{s}\frac{\ell^{2}-(x_{i}-x_{i}^{0})^{2}}{\ell^{3}} ) & \mbox{if } x\in B(x^{0},s)\setminus B(x^{0},\frac{s}{2}), \end{cases}\displaystyle \end{aligned}$$
$$ \sum_{i=1}^{N}\frac{\partial^{2} w(x)}{\partial x_{i}^{2}}= \textstyle\begin{cases} 0&\mbox{if } x \in \overline{\Omega }\setminus B(x^{0},s)\cup B(x^{0},\frac{s}{2}), \\ d ( \frac{12l(N+1)}{s^{3}}-\frac{24N}{s^{2}}+\frac{9}{s}\frac{N-1}{ \ell } ) & \mbox{if } x\in B(x^{0},s)\setminus B(x^{0},\frac{s}{2}). \end{cases} $$
$$\begin{aligned}& \int_{\Omega } \bigl\vert \Delta w(x) \bigr\vert ^{p}\,{\mathrm{d}}x=d^{p}\frac{2\pi^{\frac{N}{2}}}{\Gamma (\frac{N}{2})} \int_{\frac{s}{2}}^{s} \biggl\vert \frac{12(N+1)}{s^{3}}r- \frac{24N}{s^{2}}+\frac{9(N-1)}{s}\frac{1}{r} \biggr\vert ^{p}r^{N-1} \,{\mathrm{d}}r, \\& \int_{\Omega } \bigl\vert \nabla w(x) \bigr\vert ^{p}\,{\mathrm{d}}x \\& \quad =\int_{B(x^{0},s)\backslash B(x^{0},\frac{s}{2})} \Biggl[ \sum_{i=1}^{N}d ^{2} \biggl( \frac{12l(x_{i}-x^{0}_{i})}{s^{3}}- \frac{24(x_{i}-x^{0} _{i})}{s^{2}}+\frac{9}{s} \frac{(x_{i}-x^{0}_{i})}{l} \biggr) ^{2} \Biggr] ^{\frac{p}{2}}\,{\mathrm{d}}x \\& \quad = d^{p} \int_{B(x^{0},s)\backslash B(x^{0},\frac{s}{2})} \Biggl[ \sum_{i=1} ^{N} \biggl( \frac{12l(x_{i}-x^{0}_{i})}{s^{3}}- \frac{24(x_{i}-x^{0} _{i})}{s^{2}}+\frac{9}{s} \frac{(x_{i}-x^{0}_{i})}{l} \biggr) ^{2} \Biggr] ^{\frac{p}{2}}\,{\mathrm{d}}x \end{aligned}$$
$$ \int_{\Omega } \bigl\vert w(x) \bigr\vert ^{p}\,{\mathrm{d}}x=d^{p}\frac{2\pi^{\frac{N}{2}}}{ \Gamma (\frac{N}{2})} \biggl( \frac{(\frac{s}{2})^{N}}{N}+ \int_{ \frac{s}{2}}^{s} \biggl\vert \frac{4}{s^{3}}r^{3}- \frac{12}{s^{2}}r^{2}+\frac{9}{s}r-1 \biggr\vert ^{p}r^{N-1}\,{\mathrm{d}}r \biggr) . $$
$$\begin{aligned} \frac{d^{p}}{p}M^{-}L &\leq \frac{1}{p} \bigl( \vartheta_{1}d^{p} +m_{0}^{p-1} \vartheta_{2}d^{p}+ \varrho \vartheta_{3}d^{p} \bigr) \\ &\leq \Phi (w) \\ &= \frac{1}{p} \bigl(\vartheta_{1}d^{p} +\widehat{M} \bigl(\vartheta_{2}d^{p} \bigr) + \varrho \vartheta_{3}d^{p} \bigr) \\ & \leq \frac{1}{p} \bigl(\vartheta_{1}d^{p}+m_{1}^{p-1} \vartheta_{2}d^{p}+ \varrho \vartheta_{3}d^{p} \bigr) \\ &\leq \frac{d^{p}}{p}M^{+}L. \end{aligned}$$
(3.12)
Theorem 3.5
Assume that Assumption
\((\mathcal{B}_{1})\)
in Theorem
3.3
and Assumption
\((\mathcal{A}_{3})\)
in Theorem
3.4
hold and there exist three positive constants
c, d
and
α
with
\(\sqrt[p]{L}d< c\), where
L
is given as in (3.10), and
\(\alpha \in [0,p)\)
such that
Then, there exist a nonempty open set
\(A\subset [0,+\infty )\)
and a positive number
\(R'\)
with the following property: for every
\(\lambda \in A\)
and every continuous function
\(g:\Omega \times \mathbb{R}\to \mathbb{R}\), there exists
\(\delta >0\)
such that, for each
\(\mu \in [0,\delta ]\), the problem (\(P^{f,g}_{\lambda ,\mu}\)) has at least three weak solutions whose norms in E are less than
\(R'\).
- \((\mathcal{B}_{3})\) :
-
\(\max_{x\in \overline{\Omega },\vert u \vert \leq c}F(x,u)<\frac{M ^{-}}{M^{+}{\operatorname{meas}}(\Omega )}\int_{B(x^{0},\frac{s}{2})}F(x,d)\,\mathrm{d}x\).
Proof
We claim that all the hypotheses of Theorem 3.1 are satisfied by choosing w as given in (3.11) and \(r<\frac{LM^{-}}{p}d ^{p}\). We observe that
where \(K_{w}\) is as given in Assumption \((\mathcal{A}_{1})\). Owing to \((\mathcal{B}_{3})\) and \(F(t,0)=0\), one has \(\int_{B(x^{0}, \frac{s}{2})}F(x,w(x))\,\mathrm{d}x>0\). So, by \((\mathcal{A}_{3})\), \((\mathcal{B}_{3})\) and (3.12), we have
Thus, Assumption \((\mathcal{B}_{2})\) in Theorem 3.3 holds. Therefore, by Theorem 3.3, for each compact interval \([a,b]\subseteq (\lambda_{1},\lambda_{2})\), there exists \(R'>0\) with the following property: for every \(\lambda \in [a,b]\) and every \(\mathrm{L}^{1}\)-Carathéodory function \(g:\Omega \times \mathbb{R} \to \mathbb{R}\), there exists \(\delta >0\) such that, for each \(\mu \in [0,\delta ]\), \(\Phi '(u)-\lambda J'(u)-\mu \Psi '(u)=0\) has at least three solutions in E. Hence, the problem (\(P^{f,g}_{\lambda ,\mu}\)) has at least three weak solutions whose norms are less than \(R'\). □
$$ K_{w}=\frac{1}{p} \bigl(\vartheta_{1}d^{p} +\widehat{M} \bigl(\vartheta_{2}d^{p} \bigr)+ \varrho \vartheta_{3}d^{p} \bigr)\geq \frac{LM^{-}}{p}d^{p}>r, $$
$$\begin{aligned} r\frac{\int_{\Omega }F(x,w(x))\,\mathrm{d}x}{\operatorname{meas}(\Omega )K_{w}} &=\frac{LM ^{-}d^{p}}{p}\frac{p\int_{\Omega }F(x,w(x))\,\mathrm{d}x}{\operatorname{meas}( \Omega )(\theta_{1}d^{p} +\widehat{M}(\theta_{2}d^{p}) + \varrho \theta_{3}d^{p})} \\ &> \frac{M^{-}}{\operatorname{meas}(\Omega )M^{+}} \int_{B(x^{0},\frac{s}{2})}F(x,d) \,\mathrm{d}x \\ &> \max_{x\in \overline{\Omega },\vert u \vert \leq c}F(x,u) \\ &> \max_{x\in \overline{\Omega },\vert u \vert \leq \sqrt[p]{\frac{pr}{M^{-}}}}F(x,u). \end{aligned}$$
Remark 3.1
Clearly, Theorem 3.5 gives the result of at least three solutions for the problem (\(P^{f,g}_{\lambda ,\mu}\)) with \(F(x,u)\) being of subquadratic growth.
Remark 3.2
The statements of Theorems 3.4 and 3.5 depend upon the test function w defined by (3.11). If we take the other choices of w, we have another statement. For example, if \(x^{0} \in \Omega \) and we pick \(s>0\) such that \(B(x^{0},s)\subset \Omega \), where \(B(x^{0}, s)\) denotes the ball with center at \(x^{0}\) and radius of s, and
and where Γ denotes the gamma function,
and we take
with \(\ell =\operatorname{dist}(x,x^{0})=\sqrt{\sum_{i=1}^{N} ( x _{i}-x_{i}^{0} ) ^{2}}\) (see [37]), then we have
and
It is easy to see that \(w\in {\mathrm{E}}\) and, in particular, since
and
one has
Therefore, condition \((\mathcal{A}_{4})\) in Theorem 3.4 takes the following form:
Moreover, the condition \(\sqrt[p]{L}d< c\) in Theorem 3.5 becomes the condition \(\sqrt[p]{L'}d< c\). Also, by choosing w as given in [37], Remark 3.4, which is as follows:
where \(\ell =\operatorname{dist}(x,x^{0})=\sqrt{\sum_{i=1}^{N} ( x_{i}-x_{i}^{0} ) ^{2}}\) and \(r_{1}, r_{2}\in \mathbb{R}\) with \(r_{2}>r_{1}>0\) (see [33, 34]), we have other forms of conditions \((\mathcal{A}_{4})\) and \(\sqrt[p]{L}d< c\).
$$\begin{aligned}& \vartheta_{1}':=\frac{2^{5P+1}\pi^{N/2}d^{p}}{s^{4p}\Gamma (N/2)} \int_{s/2}^{s} \bigl\vert 2(N+2)r^{2}-3(N+1)sr+Nr^{2} \bigr\vert ^{p}r^{N+1}\,{\mathrm{d}}r, \\& \vartheta_{2}':= \biggl( \frac{32}{s^{4}} \biggr) ^{p} \int_{B(x^{0},s)\backslash B(x^{0},\frac{s}{2})} \Biggl[ \sum_{i=1}^{N} \bigl( (s-\ell ) (s-3\ell ) \bigl(x_{i}-x_{i}^{0} \bigr) \bigr) ^{2} \Biggr] ^{ \frac{p}{2}}\,{\mathrm{d}}x, \\& \vartheta_{3}':= \biggl( \frac{16}{s^{4}} \biggr) ^{p}\frac{2\pi^{ \frac{N}{2}}}{\Gamma (\frac{N}{2})} \biggl( \frac{(\frac{s}{2})^{N}}{N}+ \int_{\frac{s}{2}}^{s} \bigl\vert r^{2}(s-r) \bigr\vert ^{p}r^{N-1}\,{\mathrm{d}}r \biggr) \end{aligned}$$
$$ L':=\vartheta_{1}'+\vartheta_{2}'+ \vartheta_{3}', $$
$$ w(x):=\left \{ \textstyle\begin{array}{l@{\quad}l} 0 &\mbox{if } x \in \overline{\Omega }\setminus B(x^{0} , s), \\ 16\frac{\ell^{2}}{s^{4}}(s- \ell )^{2}d &\mbox{if } x\in B(x^{0},s) \setminus B(x^{0},\frac{s}{2}), \\ d &\mbox{if } x \in B(x^{0},\frac{ s}{2}) \end{array}\displaystyle \right . $$
(3.13)
$$\begin{aligned}& \frac{\partial w(x)}{\partial x_{i}}=\left \{ \textstyle\begin{array}{l@{\quad}l} 0&\mbox{if } x \in \overline{\Omega }\setminus B(x^{0},s)\cup B(x^{0},\frac{s}{2}), \\ \frac{32d}{s^{4}}(s-\ell )(s-3\ell )(x_{i}-x_{i}^{0})& \mbox{if } x\in B(x^{0},s)\setminus B(x^{0},\frac{s}{2}), \end{array}\displaystyle \right . \\& \frac{\partial^{2} w(x)}{\partial x_{i}^{2}}= \left \{ \textstyle\begin{array}{l@{\quad}l} 0&\mbox{if } x \in \overline{\Omega }\setminus B(x^{0},s)\cup B(x^{0},\frac{s}{2}), \\ \frac{32(x_{i}-x_{i}^{0})d}{s^{4}} ( \frac{6(x_{i}-x_{i}^{0})( \ell -4s)}{\ell }+s^{2}-4ls+3l^{2} ) & \mbox{if } x\in B(x^{0},s)\setminus B(x^{0},\frac{s}{2}) \end{array}\displaystyle \right . \end{aligned}$$
$$ \sum_{i=1}^{N}\frac{\partial^{2} w(x)}{\partial x_{i}^{2}}=\left \{ \textstyle\begin{array}{l@{\quad}l} 0&\mbox{if } x \in \overline{\Omega }\setminus B(x^{0},s)\cup B(x^{0},\frac{s}{2}), \\ 32d ( \frac{2(N+2)\ell^{2}-3s(N+1)\ell +Ns^{2}}{s^{4}} ) & \mbox{if } x\in B(x^{0},s)\setminus B(x^{0},\frac{s}{2}). \end{array}\displaystyle \right . $$
$$\begin{aligned} &\int_{\Omega } \bigl\vert \Delta w(x) \bigr\vert ^{p}\,{\mathrm{d}}x =\frac{2^{5P+1}\pi^{N/2}d^{p}}{s^{4p}\Gamma (N/2)}\int_{s/2}^{s} \bigl\vert 2(N+2)r^{2}-3(N+1)rs+Nr^{2}\bigr\vert ^{p}r^{N+1}\,{\mathrm{d}}r, \\ &\int_{\Omega } \bigl\vert \nabla w(x) \bigr\vert ^{p}\,{\mathrm{d}}x \\ &\quad = \int_{B(x^{0},s)\backslash B(x^{0},\frac{s}{2})} \Biggl[ \sum_{i=1}^{N} \biggl( \frac{32d}{s^{4}}(s-\ell ) (s-3\ell ) \bigl(x_{i}-x_{i}^{0} \bigr) \biggr) ^{2} \Biggr] ^{\frac{p}{2}}\,{\mathrm{d}}x \\ &\quad = \biggl( \frac{32d}{s^{4}} \biggr) ^{p} \int_{B(x^{0},s)\backslash B(x^{0},\frac{s}{2})} \Biggl[ \sum_{i=1}^{N} \bigl( (s-\ell ) (s-3\ell ) \bigl(x_{i}-x_{i}^{0} \bigr) \bigr) ^{2} \Biggr] ^{ \frac{p}{2}}\,{\mathrm{d}}x \end{aligned}$$
$$ \int_{\Omega } \bigl\vert w(x) \bigr\vert ^{p}\,{ \mathrm{d}}x= \biggl( \frac{16d}{s^{4}} \biggr) ^{p}\frac{2 \pi^{\frac{N}{2}}}{\Gamma (\frac{N}{2})} \biggl( \frac{(\frac{s}{2})^{N}}{N}+ \int_{\frac{s}{2}}^{s} \bigl\vert r^{2}(s-r) \bigr\vert ^{p}r^{N-1}\,{\mathrm{d}}r \biggr) , $$
$$\begin{aligned} \frac{d^{p}}{p}M^{-}L'&\leq \frac{1}{p} \bigl( \vartheta_{1}'d^{p} +m_{0}^{p-1} \vartheta_{2}'d^{p}+ \varrho \vartheta_{3}'d^{p} \bigr) \\ &\leq \Phi (w) \\ &= \frac{1}{p} \bigl(\vartheta_{1}'d^{p} + \widehat{M} \bigl(\vartheta_{2}'d^{p} \bigr) + \varrho \vartheta_{3}'d^{p} \bigr) \\ & \leq \frac{1}{p} \bigl(\vartheta_{1}'d^{p}+m_{1}^{p-1} \vartheta_{2}'d^{p}+ \varrho \vartheta_{3}'d^{p} \bigr) \\ &\leq \frac{d^{p}}{p}M^{+}L'. \end{aligned}$$
- \(({\mathcal{A}_{5}})\) :
-
there exists a positive constant d such thatwhere w is given by (3.13).$$ \vartheta_{1}'d^{p} +\widehat{M} \bigl( \vartheta_{2}'d^{p} \bigr) +\varrho \vartheta_{3}'d^{p}\neq 0\quad\mbox{and}\quad \varepsilon < \frac{pM ^{-}\int_{\Omega }F(x,w(x))\,\mathrm{d}x}{k^{p}(\vartheta_{1}'d^{p}+ \widehat{M}(\vartheta_{2}'d^{p})+\varrho \vartheta_{3}'d^{p})}, $$
$$ w(x):=\left \{ \textstyle\begin{array}{l@{\quad}l} 0 &\mbox{if } x \in \overline{\Omega }\setminus B(x^{0},r_{2}), \\ \frac{3(\ell^{4}-r^{4})-4(r_{1}+r_{2})(\ell^{3}-r_{2}^{3})+6r_{1}r _{2}(\ell^{2}-r_{2}^{2})}{(r_{2}-r_{1})^{3}(r_{1}+r_{2})}d & \mbox{if } x\in B(x^{0},r_{2})\setminus B(x^{0},r_{1}), \\ d &\mbox{if } x \in B(x^{0},r_{1}), \end{array}\displaystyle \right . $$
Now, we point out some results in which the function f has separated variables. To be precise, consider the following problem:
where \(\theta :\Omega \to \mathbb{R}\) is a nonnegative and nonzero function, \(\theta \in {\mathrm{L}}^{1}(\Omega )\), \(f:\mathbb{R}\to \mathbb{R}\) is a continuous function and \(g:\Omega \times \mathbb{R} \to \mathbb{R}\) is as introduced in the problem (\(P^{f,g}_{\lambda ,\mu}\)) in Introduction.
$$ \left \{ \textstyle\begin{array}{l@{\quad}l} T(u)= \lambda \theta (x)f(u)+\mu g(x,u),&\textrm{in }\Omega , \\ u=\Delta u=0,&\textrm{on } \partial \Omega , \end{array}\displaystyle \right . \hspace{30pt} (P^{f,\theta ,g}_{\lambda ,\mu }) $$
Set \(F(x,t)= \theta (x)F(t)\) for every \((x,t)\in \Omega \times \mathbb{R}\), where
for all \(t\in \mathbb{R}\). The following existence results are consequences of Theorem 3.4.
$$ F(t)= \int_{0}^{t} f(\xi )\,\mathrm{d}\xi $$
Theorem 3.6
Assume that
Then, for each compact interval
\([c,d]\subset (\lambda_{3},\lambda _{4})\), where
\(\lambda_{3}\)
and
\(\lambda_{4}\)
are the same as
\(\lambda_{1}\)
and
\(\lambda_{2}\), but
\(\int_{\Omega }F(x,u(x))\,\mathrm{d}x\)
is replaced by
\(\int_{\Omega } \theta (x)F(u(x))\,\mathrm{d}x\), respectively, there exists
\(R> 0\)
with the following property: for every
\(\lambda \in [c,d]\)
and every
\(\mathrm{L}^{1}\)-Carathéodory function
\(g:\Omega \times \mathbb{R}\to \mathbb{R}\), there exists
\(\gamma >0\)
such that for each
\(\mu \in [0,\gamma ]\), the problem (\(P^{f,\theta ,g}_{\lambda ,\mu }\)) has at least three weak solutions whose norms in E are less than
R.
- \((\mathcal{A}_{6})\) :
-
there exists a constant \(\varepsilon >0\) such that$$ \sup_{x\in \Omega }\theta (x)\cdot\max \biggl\{ \limsup _{u\rightarrow 0} \frac{F(u)}{\vert u \vert ^{p}} ,\,\limsup_{\vert u \vert \rightarrow \infty } \frac{F(u)}{\vert u \vert ^{p}} \biggr\} < \varepsilon ; $$
- \((\mathcal{A}_{7})\) :
-
there exists a positive constant d such thatwhere w is given by (3.11).$$ \vartheta_{1}d^{p} +\widehat{M} \bigl(\vartheta_{2}d^{p} \bigr) +\varrho \vartheta _{3}d^{p}\neq 0\quad \textit{and }\quad \varepsilon < \frac{pM^{-} \int_{\Omega }F(x,w(x))\,\mathrm{d}x}{k^{p}(\vartheta_{1}d^{p}+\widehat{M}( \vartheta_{2}d^{p})+\varrho \vartheta_{3}d^{p})}, $$
Theorem 3.7
Assume that there exists a positive constant
d
such that
Moreover, suppose that
Then, for each compact interval
\([c,d]\subset (\lambda_{3},\infty )\), where
\(\lambda_{3}\)
is the same as
\(\lambda_{1}\)
but
\(\int_{\Omega }F(x, u(x)) \,\mathrm{d}x\)
is replaced by
\(\int_{\Omega } \theta (x)F(u(x))\,\mathrm{d}x\), there exists
\(R>0\)
with the following property: for every
\(\lambda \in [c,d]\)
and every
\(\mathrm{L}^{1}\)-Carathéodory function
\(g:\Omega \times \mathbb{R}\to \mathbb{R}\), there exists
\(\gamma >0\)
such that for each
\(\mu \in [0,\gamma ]\), the problem (\(P^{f,\theta ,g}_{\lambda ,\mu }\)) has at least three weak solutions whose norms in E are less than R.
$$ \vartheta_{1}d^{p} +\widehat{M} \bigl( \vartheta_{2}d^{p} \bigr)+\varrho \vartheta _{3}d^{p}>0\quad \textit{and}\quad F(d)>0. $$
(3.14)
$$ \limsup_{u\rightarrow 0}\frac{f(u)}{\vert u \vert ^{p-1}}= \limsup _{\vert u \vert \rightarrow \infty }\frac{f(u)}{\vert u \vert ^{p-1}}=0. $$
(3.15)
Proof
We easily observe that from (3.15) Assumption \((\mathcal{A} _{6})\) is satisfied for every \(\varepsilon >0\). Moreover, using (3.14), by choosing \(\varepsilon >0\) small enough, one can derive Assumption \((\mathcal{A}_{7})\). Hence, the conclusion follows from Theorem 3.6. □
Remark 3.3
Our results show that no asymptotic conditions on f and g are required, and merely the algebraic conditions on f are supposed to guarantee the existence of solutions.
Now, we present the following example to illustrate Theorem 3.7.
Example 3.1
Let \(N=2\), \(p=6\), \(\varrho =2\), \(\Omega =\{(x_{1},x_{2})\in \mathbb{R}^{2};\, x_{1}^{2}+x_{2}^{2}<5\}\subset \mathbb{R}^{2}\), \(x^{0}=(0,0)\), \(s=2\),
\(\vartheta (x)=1\) for all \(x\in \Omega \) and
Thus,
\(m_{0}=1\), \(m_{1}=2\) and f is a nonnegative and continuous function. By choosing \(d=1\), we have
Therefore,
and
are positive. So,
Moreover, we have
and
Hence, by applying Theorem 3.7 for each compact interval \([c,d]\subset (0,\infty )\), there exists \(R>0\) with the following property: for every \(\lambda \in [c,d]\) and every continuous function \(g:\mathbb{R}\to \mathbb{R}\), there exists \(\gamma >0\) such that, for each \(\mu \in [0,\gamma ]\), the problem
has at least three weak solutions whose norms in the space \(\mathrm{W} ^{2,6}(\Omega )\cap {\mathrm{W}}^{1,6}_{0}(\Omega )\) are less than R.
$$ M(t)=1+\frac{1}{\cosh t}\quad \mbox{for all } t\in \mathbb{R}, $$
$$ f(t)=\left \{ \textstyle\begin{array}{l@{\quad}l} 2(t+\sin t)^{2},& \text{if} \,\, t< \pi , \\ 2\pi^{2}+\tanh (t-\pi ),& \text{if} \,\, t\geq \pi . \end{array}\displaystyle \right . $$
$$ S \bigl(x^{0},s \bigr)=S \bigl((0,0),2 \bigr)= \bigl\{ (x_{1},x_{2}) \in \mathbb{R}^{2};\, x_{1}^{2}+x _{2}^{2}< 4 \bigr\} \subset \Omega , $$
$$ w(x_{1},x_{2})= \textstyle\begin{cases} 0&\mbox{if } (x_{1},x_{2})\in \overline{\Omega }\setminus S ((0,0),2 ), \\ 1&\mbox{if } (x_{1},x_{2})\in S ((0,0),1 ), \\ \frac{1}{2}\sqrt{(x_{1}^{2}+x_{2}^{2})^{3}} \\ \quad {}-3(x_{1}^{2}+x_{2}^{2}) + \frac{9}{2}\sqrt{x_{1}^{2}+x_{2}^{2}}-1&\mbox{if } (x_{1},x_{2})\in S ((0,0),2 )\setminus S ((0,0),1 ). \end{cases} $$
$$\begin{aligned}& \vartheta_{1}=2\pi \int_{1}^{2} \biggl( \frac{9}{2}\xi -12+ \frac{9}{2\xi ^{3}} \biggr) ^{6}\xi \,{\mathrm{d}}\xi , \\& \vartheta_{2}= \iint_{S ((0,0),2 )\setminus S ((0,0),1 )} \Biggl[\sum_{i=1}^{2} \biggl(\frac{3}{2}x_{i}\sqrt{x_{1}^{2}+x_{2}^{2}}-6x _{i}+\frac{9x_{i}}{2\sqrt{x_{1}^{2}+x_{2}^{2}}} \biggr)^{2} \Biggr]^{4} { \mathrm{d}}x_{1}\,{\mathrm{d}}x_{2} \end{aligned}$$
$$ \vartheta_{3}=2\pi \biggl( \frac{1}{2}+ \int_{1}^{2} \biggl( \frac{1}{2}\xi ^{3}-3\xi^{2}+\frac{9}{2}\xi -1 \biggr) ^{6}\xi \,{\mathrm{d}}\xi \biggr) $$
$$ \vartheta_{1}d^{p} +\widehat{M} \bigl(\vartheta_{2}d^{p} \bigr) +\varrho \vartheta _{3}d^{p}>0. $$
$$\begin{aligned} F(d)=F(1)= \int_{0}^{1} 2(t+\sin t)^{2}\,{\mathrm{d}}t>0 \end{aligned}$$
$$ \lim_{u\to 0}\frac{f(u)}{\vert u \vert }=\lim_{u\rightarrow \infty } \frac{f(u)}{\vert u \vert }=0. $$
$$ \left \{ \textstyle\begin{array}{l@{\quad}l} \Delta (\vert \Delta u \vert ^{4}\Delta u)- (1+\frac{1}{\cosh (\int_{\Omega }\vert \nabla u \vert ^{6}\,{\mathrm{d}}x)} )^{5}\Delta_{6}u+2\vert u \vert ^{4}u=\lambda f(u)+ \mu g(u)&\mbox{in } \Omega , \\ u=\Delta u=0 &\mbox{on } \partial \Omega \end{array}\displaystyle \right . $$
The following existence result is a consequence of Theorem 3.5.
Theorem 3.8
Assume that there exist five positive constants
c, d, α, ν
and
ξ
with
\(c>\sqrt[p]{L}d\), where
L
is given by (3.10), and
\(\alpha \in [0,p)\)
such that
Then there exist a nonempty open set
\(A\subset [0,\infty )\)
and a positive number
\(R'\)
with the following property: for every
\(\lambda \in A\)
and every continuous function
\(g:\mathbb{R}\to \mathbb{R}\), there exists
\(\delta >0\)
such that, for each
\(\mu \in [0, \delta ]\), the problem (\(P^{f,\theta ,g}_{\lambda ,\mu }\)) has at least three weak solutions whose norms in E are less than
\(R'\).
- \((\mathcal{B}_{4})\) :
-
\(\max_{\vert u \vert \leq c}F(u)<\frac{M^{-}F(d)\pi ^{N/2}}{M^{+}{\operatorname{meas}}(\Omega )\Gamma (\frac{N}{2})}\);
- \((\mathcal{B}_{5})\) :
-
\(F(u)>0\) for each \(u\in \mathbb{R}\);
- \((\mathcal{B}_{6})\) :
-
\(\vert F(u) \vert \leq \nu \vert u \vert ^{\alpha }+\xi \) for all \(u\in \mathbb{R}\).
Finally, we present the following example to illustrate Theorem 3.8.
Example 3.2
Let \(N=2\), \(p=3\), \(\varrho =1\), \(\Omega =\{(x_{1},x_{2})\in \mathbb{R}^{2};\, x_{1}^{2}+x_{2}^{2}<\frac{1}{100}\}\subset \mathbb{R}^{2}\), \(x^{0}=(0,0)\), \(s=\frac{1}{20}\),
and
Thus,
\(m_{0}=1\), \(m_{1}=3\), and f is a nonnegative and continuous function. By choosing \(c=100\), \(d=1\), \(\alpha =2\), \(\nu =1\) and \(\xi =\pi \), we have \(\alpha =2\in [0,3)=[0,p)\), \(M^{-}=1\), \(M^{+}=3\) and \(L=58.18309 \pi \). Therefore,
and
Hence, by applying Theorem 3.8, there exist a nonempty open set \(A\subset [0,+\infty )\) and a positive number \(R'\) with the following property: for every \(\lambda \in A\) and every nonnegative continuous function \(g:\Omega \times \mathbb{R}\to \mathbb{R}\), there exists \(\delta >0\) such that, for each \(\mu \in [0,\delta ]\), the problem
has at least three weak solutions whose norms in the space \(\mathrm{W} ^{2,3}(\Omega )\cap {\mathrm{W}}^{1,3}_{0}(\Omega )\) are less than R.
$$ M(t)=2+\tanh t\quad \mbox{for all } t\in [0,+\infty [ $$
$$ f(t)=\frac{1}{1+t^{2}}\quad \mbox{for all } \, t\in \mathbb{R}. $$
$$ S \bigl(x^{0},s \bigr)=S \biggl((0,0),\frac{1}{20} \biggr)= \biggl\{ (x_{1},x_{2})\in \mathbb{R}^{2};\, x_{1}^{2}+x_{2}^{2}< \frac{1}{400} \biggr\} \subset \Omega , $$
$$\begin{aligned}& c=100>\sqrt[3]{58.18309\pi }>\sqrt[p]{L}d, \\& \max_{\vert u \vert \leq c}F(u)=\max_{\vert u \vert \leq 100}F(u)=\arctan (100)< \frac{ \pi }{2}< \frac{100\pi }{36}=\frac{M^{-}F(d)\pi }{M^{+}{\operatorname{meas}}( \Omega )}, \\& F(d)=F(1)=\arctan (1)=\frac{\pi }{4}>0 \end{aligned}$$
$$ \bigl\vert F(u) \bigr\vert \leq \vert u \vert ^{2}+\pi \quad \mbox{for all } u\in \mathbb{R}. $$
$$ \left \{ \textstyle\begin{array}{l@{\quad}l} \Delta (\vert \Delta u \vert \Delta u)- (2+\tanh (\int_{\Omega }\vert \nabla u \vert ^{3} \,{\mathrm{d}}x) )^{2}\Delta_{3}u+\vert u \vert u=\lambda \frac{1}{1+u^{2}}+\mu g(u)& \mbox{in }\Omega , \\ u=\Delta u=0 &\mbox{on } \partial \Omega \end{array}\displaystyle \right . $$
Remark 3.4
We point out that the same statements of the above given results can be obtained by considering the special case
where \(b_{1}\), \(b_{2}\), ι and κ are positive numbers. In fact, we have
Arguing as in the proof of Theorems 3.1 and 3.5, three weak solutions can be obtained.
$$ M(t)= b_{1}+b_{2}t \quad \mbox{for } t\in [\iota ,\kappa ], $$
$$\begin{aligned}& \widehat{M}(t)= \int_{0}^{t} (b_{1}+b_{2}\xi ) \,\mathrm{d}\xi = \frac{(b _{1}+b_{2}t)^{2}}{2b_{2}}-\frac{b_{1}^{2}}{2b_{2}} \quad \textrm{for } t\in [\iota ,\kappa ], \\& m_{1}=b_{1}+b_{2}\iota \quad \mbox{and}\quad m_{2}=b_{1}+b_{2}\kappa . \end{aligned}$$
4 Concluding remarks
Kirchhoff’s model as an extension of the classical D’Alembert’s wave equation for free vibrations of elastic strings takes into account the changes in length of the string produced by transverse vibrations. It received great attention only after Lions had proposed an abstract framework for the problem. On the other hand, fourth-order boundary value problems which describe the deformations of an elastic beam in an equilibrium state whose both ends are simply supported have been extensively studied in the literature. Here we have investigated the existence of multiple solutions for perturbed nonlocal fourth-order equations of Kirchhoff type under Navier boundary conditions. We have given some new criteria for guaranteeing that the perturbed fourth-order equations of Kirchhoff type possess at least three weak solutions by using a variational method and some critical point theorems due to Ricceri.
Declarations
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
(1)
Department of Mathematics, Faculty of Sciences, Razi University, 67149, Kermanshah, Iran
(2)
Young researchers and elite club, Kermanshah branch, Islamic Azad University, Kermanshah, Iran
(3)
Department of Economics, University of Messina, via dei Verdi 75, Messina, Italy
References
- Kirchhoff, G: Vorlesungen uber mathematische Physik: Mechanik. Teubner, Leipzig (1883) MATHGoogle Scholar
- Shahruz, SM, Parasurama, SA: Suppression of vibration in the axially moving Kirchhoff string by boundary control. J. Sound Vib. 214, 567-575 (1998) MathSciNetView ArticleMATHGoogle Scholar
- Alves, CO, Corrêa, FSJA, Ma, TF: Positive solutions for quasilinear elliptic equations of Kirchhoff type. Comput. Math. Appl. 49, 85-93 (2005) MathSciNetView ArticleMATHGoogle Scholar
- Chipot, M, Lovat, B: Some remarks on non local elliptic and parabolic problems. Nonlinear Anal. TMA 30, 4619-4627 (1997) View ArticleMATHGoogle Scholar
- Graef, JR, Heidarkhani, S, Kong, L: A variational approach to a Kirchhoff-type problem involving two parameters. Results Math. 63, 877-889 (2013) MathSciNetView ArticleMATHGoogle Scholar
- He, X, Zou, W: Infinitely many positive solutions for Kirchhoff-type problems. Nonlinear Anal. TMA 70, 1407-1414 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Ma, TF: Remarks on an elliptic equation of Kirchhoff type. Nonlinear Anal. TMA 63, 1967-1977 (2005) View ArticleGoogle Scholar
- Mao, A, Zhang, Z: Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition. Nonlinear Anal. TMA 70, 1275-1287 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Molica Bisci, G, Rădulescu, V: Applications of local linking to nonlocal Neumann problems. Commun. Contemp. Math. 17, 1450001 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Molica Bisci, G, Rădulescu, V: Mountain pass solutions for nonlocal equations. Ann. Acad. Sci. Fenn., Math. 39, 579-592 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Perera, K, Zhang, ZT: Nontrivial solutions of Kirchhoff-type problems via the Yang index. J. Differ. Equ. 221, 246-255 (2006) MathSciNetView ArticleMATHGoogle Scholar
- Ricceri, B: On an elliptic Kirchhoff-type problem depending on two parameters. J. Glob. Optim. 46, 543-549 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Colasuonno, F, Pucci, P: Multiplicity of solutions for \(p(x)\)-polyharmonic elliptic Kirchhoff equations. Nonlinear Anal. TMA 74, 5962-5974 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Autuori, G, Colasuonno, F, Pucci, P: On the existence of stationary solutions for higher-order p-Kirchhoff problems. Commun. Contemp. Math. 16, 1450002 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Autuori, G, Colasuonno, F, Pucci, P: Blow up at infinity of solutions of polyharmonic Kirchhoff systems. Complex Var. Elliptic Equ. 57, 379-395 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Autuori, G, Colasuonno, F, Pucci, P: Lifespan estimates for solutions of polyharmonic Kirchhoff systems. Math. Models Methods Appl. Sci. 22, 1150009 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Barrios, B, Peral, I, Vita, S: Some remarks about the summability of nonlocal nonlinear problems. Adv. Nonlinear Anal. 4, 91-107 (2015) MathSciNetMATHGoogle Scholar
- Figueiredo, GM, Molica Bisci, G, Servadei, R: On a fractional Kirchhoff-type equation via Krasnoselskii’s genus. Asymptot. Anal. 94, 347-361 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Mingqi, X, Molica Bisci, G, Tian, G, Zhang, B: Infinitely many solutions for the stationary Kirchhoff problems involving the fractional p-Laplacian. Nonlinearity 29, 357-374 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Molica Bisci, G, Rădulescu, V, Servadei, R: Variational Methods for Nonlocal Fractional Problems. Encyclopedia of Mathematics and Its Applications, vol. 162. Cambridge University Press, Cambridge (2016) View ArticleMATHGoogle Scholar
- Molica Bisci, G, Repovš, D: Existence and localization of solutions for nonlocal fractional equations. Asymptot. Anal. 90, 367-378 (2014) MathSciNetMATHGoogle Scholar
- Molica Bisci, G, Repovš, D: On doubly nonlocal fractional elliptic equations. Atti Accad. Naz. Lincei, Rend. Lincei, Mat. Appl. 26, 161-176 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Molica Bisci, G, Pizzimenti, P: Sequences of weak solutions for non-local elliptic problems with Dirichlet boundary condition. Proc. Edinb. Math. Soc. 57, 779-809 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Molica Bisci, G, Tulone, F: An existence result for fractional Kirchhoff-type equations. Z. Anal. Anwend. 35, 181-197 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Molica Bisci, G, Vilasi, L: On a fractional degenerate Kirchhoff-type problem. Commun. Contemp. Math. 1, 1550088 (2017) MathSciNetView ArticleMATHGoogle Scholar
- Pucci, P, Xiang, M, Zhang, B: Existence and multiplicity of entire solutions for fractional p-Kirchhoff equations. Adv. Nonlinear Anal. 5, 27-55 (2016) MathSciNetMATHGoogle Scholar
- Rădulescu, V, Xiang, M, Zhang, B: Existence of solutions for a bi-nonlocal fractional p-Kirchhoff type problem. Comput. Math. Appl. 71, 255-266 (2016) MathSciNetView ArticleGoogle Scholar
- Rădulescu, V, Xiang, M, Zhang, B: Multiplicity of solutions for a class of quasilinear Kirchhoff system involving the fractional p-Laplacian. Nonlinearity 29, 3186-3205 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Cabada, A, Cid, JA, Sanchez, L: Positivity and lower and upper solutions for fourth-order boundary value problems. Nonlinear Anal. TMA 67, 1599-1612 (2007) MathSciNetView ArticleMATHGoogle Scholar
- Graef, JR, Heidarkhani, S, Kong, L: Multiple solutions for a class of \((p_{1},\ldots ,p_{n})\)-biharmonic systems. Commun. Pure Appl. Anal. 12, 1393-1406 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Heidarkhani, S: Existence of non-trivial solutions for systems of n fourth order partial differential equations. Math. Slovaca 64, 1249-1266 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Heidarkhani, S, Tian, Y, Tang, C-L: Existence of three solutions for a class of \((p_{1},\ldots ,p_{n})\)-biharmonic systems with Navier boundary conditions. Ann. Pol. Math. 104, 261-277 (2012) View ArticleMATHGoogle Scholar
- Li, L, Tang, C-L: Existence of three solutions for \((p,q)\)-biharmonic systems. Nonlinear Anal. TMA 73, 796-805 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Li, C, Tang, C-L: Three solutions for a Navier boundary value problem involving the p-biharmonic. Nonlinear Anal. TMA 72, 1339-1347 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Liu, J, Chen, SX, Wu, X: Existence and multiplicity of solutions for a class of fourth-order elliptic equations in \(\mathbb{R}^{N}\). J. Math. Anal. Appl. 395, 608-615 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Liu, H, Su, N: Existence of three solutions for a p-biharmonic problem. Dyn. Contin. Discrete Impuls. Syst. 15, 445-452 (2008) MathSciNetMATHGoogle Scholar
- Molica Bisci, G, Repovš, D: Multiple solutions of p-biharmonic equations with Navier boundary conditions. Complex Var. Elliptic Equ. 59, 271-284 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Ferrara, M, Khademloo, S, Heidarkhani, S: Multiplicity results for perturbed fourth-order Kirchhoff type elliptic problems. Appl. Math. Comput. 234, 316-325 (2014) MathSciNetMATHGoogle Scholar
- Heidarkhani, S, Khademloo, S, Solimaninia, A: Multiple solutions for a perturbed fourth-order Kirchhoff type elliptic problem. Port. Math. 71(1), 39-61 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Massar, M, Hssini, EM, Tsouli, N, Talbi, M: Infinitely many solutions for a fourth-order Kirchhoff type elliptic problem. J. Math. Comput. Sci. 8, 33-51 (2014) MATHGoogle Scholar
- Wang, F, An, Y: Existence and multiplicity of solutions for a fourth-order elliptic equation. Bound. Value Probl. 2012, 6 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Wang, F, Avci, M, An, Y: Existence of solutions for fourth order elliptic equations of Kirchhoff-type. J. Math. Anal. Appl. 409, 140-146 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Ricceri, B: A further three critical points theorem. Nonlinear Anal. TMA 71, 4151-4157 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Ricceri, B: A three critical points theorem revisited. Nonlinear Anal. TMA 70, 3084-3089 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Pucci, P, Serrin, J: A mountain pass theorem. J. Differ. Equ. 60, 142-149 (1985) MathSciNetView ArticleMATHGoogle Scholar
- Pucci, P, Serrin, J: Extensions of the mountain pass theorem. J. Funct. Anal. 59, 185-210 (1984) MathSciNetView ArticleMATHGoogle Scholar
- Ricceri, B: Existence of three solutions for a class of elliptic eigenvalue problem. Math. Comput. Model. 32, 1485-1494 (2000) MathSciNetView ArticleMATHGoogle Scholar
- Chen, H, He, Z: New results for perturbed Hamiltonian systems with impulses. Appl. Math. Comput. 218, 9489-9497 (2012) MathSciNetMATHGoogle Scholar
- Sun, J, Chen, H, Nieto, JJ, Otero-Novoa, M: The multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive effects. Nonlinear Anal. TMA 72, 4575-4586 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Heidarkhani, S, Caristi, G, Salari, A: Perturbed Kirchhoff-type p-Laplacian discrete problems. Collect. Math. (2016). doi:10.1007/s13348-016-0180-4 MATHGoogle Scholar
- Heidarkhani, S, Salari, A: Existence of three solutions for impulsive perturbed elastic beam fourth-order equations of Kirchhoff-type. Studia Sci. Math. Hung. 54(1), 119-140 (2017) Google Scholar
Copyright
© The Author(s) 2017
