Skip to main content

Multiplicity results involving p-biharmonic Kirchhoff-type problems


This paper deals with the existence of multiple solutions for the following Kirchhoff type equations involving p-biharmonic operator:

$$\begin{aligned}& -M \biggl( \int_{\varOmega} \bigl( \vert \Delta_{p}u \vert ^{2}+ \vert u \vert ^{p} \bigr)\,dx \biggr) \bigl( \Delta _{p}^{2}u- \vert u \vert ^{p-2}u \bigr) =\lambda f(x) \vert u \vert ^{q-2}u+g(x) \vert u \vert ^{m-2}u,\quad x\in\varOmega, \end{aligned}$$

where Ω is a bounded domain in \(\mathbb{R}^{N}\) (\(N>1\)), \(\lambda >0\), \(p, q, m>1\), M is a continuous function, and the weight functions f and g are measurable. We obtain the existence results by combining the variational method with Nehari manifold and fibering maps.

1 Introduction

The theory of p-Laplacian and p-biharmonic operators has been developed very quickly. The investigation of the existence and multiplicity of solutions has attracted a considerable attention of researchers (see, for instance, [1, 3, 15, 18, 22, 24, 2628] and the references therein). The motivation of this interest stems from the fact that these nonhomogeneous differential operators are a very productive and rich area of research in recent decades. This theory have relevant applications in various fields; we refer the reader to [17, 2023].

Kirchhof-type equations, known as nonlocal differential equations, have received specific attention in recent years. An important number of surveys dealing with this type of equations can model phenomenons arising from the study of elastic mechanics, in numerous physical phenomena such as systems of particles in thermodynamical equilibrium, dielectric breakdown, image restoration, biological phenomena, and so on (see [9, 14, 19, 25, 30] and references therein for discussions of various applications ).

In recent years, several authors have considered the Nehari manifold to study problems involving sign-changing weight functions [2, 4, 6, 7, 1013, 15, 16, 26, 28]. More precisely, Ji and Wang [16] proved the existence of two nontrivial solutions for the following perturbed nonlinear p-biharmonic boundary value problem:

$$\left \{ \textstyle\begin{array}{l} \Delta_{p}^{2}u = \vert u \vert ^{q-2}u+\lambda h(x) \vert u \vert ^{r-2}u,\quad x\in\varOmega , \\ u=\nabla u=0 \quad\mbox{on } \partial\varOmega, \end{array}\displaystyle \right . $$

where \(1< r< p< q< p^{*}\) with \(p^{*}=\frac{Np}{N-2p}\) if \(p<\frac{N}{2}\) and \(p^{*}=\infty\) if \(p\geq\frac{N}{2}\), h is a continuous function in Ω̅, which can change sign, and \(\Delta _{p}^{2}u:=\Delta ( \vert \Delta u \vert ^{p-2}\Delta u )\) is the p-biharmonic operator.

Chen et al. [8] considered the following nonhomogeneous Kirchhof-type problem:

$$ \left \{ \textstyle\begin{array}{l} -M (\int_{\varOmega} \vert \nabla u \vert ^{2}\,dx )\Delta u =\lambda f(x) \vert u \vert ^{q-2}u+g(x) \vert u \vert ^{m-2}u,\quad x\in\varOmega , \\ u=0 \quad\text{on } \partial\varOmega, \end{array}\displaystyle \right . $$

where \(1< q<2<m<2^{*}\) (\(2^{*}=\frac{2N}{N-2}\) if \(N\geq3\), \(2^{*}=\infty\) if \(N=1,2\)), \(M(s)=a+bs\), and a, b, λ, are positive real numbers. The weight functions f and g are continuous in Ω̅. Based on the Nehari manifold method and the fibering maps, the authors proved that problem (1.1) admits at least two nontrivial solutions.

Inspired by the works mentioned, we study the following Kirchhof-type system:

$$ \left \{ \textstyle\begin{array}{l} -M (\int_{\varOmega}( \vert \Delta_{p}u \vert ^{2}+ \vert u \vert ^{p})\,dx ) (\Delta_{p}^{2}u- \vert u \vert ^{p-2}u)\\\quad=\lambda f(x) \vert u \vert ^{q-2}u+g(x) \vert u \vert ^{m-2}u,\quad x\in\varOmega,\\ u\in W^{2,p}(\varOmega)\setminus\{0\}, \end{array}\displaystyle \right . $$

where \(\varOmega\subset\mathbb{R}^{N}\) (\(N\geq2\)) is a bounded domain with smooth boundary ∂Ω, \(\lambda>0\), the functions f, g are measurable in Ω, and the function M is defined on \([0,\infty)\) by \(M(s)=a+b s^{l}\) for some \(a,b >0\) and \(0\leq l<\frac {2p}{N-2p}\).

Before giving our main result, we assume the following hypotheses:


g is a measurable function such that \(g\in L^{\frac {p^{*}}{p^{*}-m}}(\varOmega)\) and \(g^{+}:=\max(g,0)\neq0\).


f is a measurable function such that \(f\in L^{\frac {p^{*}}{p^{*}-q}}(\varOmega)\) and \(f^{+}:=\max(f,0)\neq0\).

Our main result of this paper is the following theorem.

Theorem 1.1

Assume\((H_{1})\)\((H_{2})\). If\(2p< N\)and\(1< m< p\leq p(l+1)<q<p^{*}\), then there exists\(\lambda_{0}>0\)such that for all\(|\lambda|\in(0,\lambda_{0})\), problem (1.2) has at least two nontrivial solutions.

The rest of this paper is organized as follows. In Sect. 2, we give some definitions and basic results that will be used in this paper. Section 3 is devoted to the proof of Theorem 1.1.

2 Definitions and basic results

In this section, we collect some basic preliminary results that will be used in the proof of our main result. To state our main result, let us introduce some definitions and notations. First, we define the Sobolev space

$$W^{2,p}(\varOmega)= \bigl\{ u\in L^{p}(\varOmega), \vert \Delta u \vert \in L^{p}(\varOmega ) \bigr\} $$

equipped with the norm

$$\Vert u \Vert = \biggl( \int_{\varOmega} \bigl( \vert \Delta u \vert ^{p}+ \vert u \vert ^{p} \bigr)\,dx \biggr)^{\frac{1}{p}}. $$

For \(1< s\leq p^{*}\), we denote by \(C_{s}\) the best Sobolev constant for the embedding operator \(W^{2,p}(\varOmega)\hookrightarrow L^{s}(\varOmega )\), which is given by

$$\begin{aligned} C_{s}:=\inf_{u\in W^{2,p}(\varOmega)\setminus\{0\}}\frac{\int_{\varOmega } \vert \Delta u \vert ^{p}\,dx}{ (\int_{\varOmega} \vert u \vert ^{s}\,dx )^{\frac{p}{s}}}. \end{aligned}$$

In particular, we have

$$\biggl( \int_{\varOmega} \vert u \vert ^{s}\,dx \biggr)^{\frac{1}{s}}\leq(C_{s})^{-\frac {1}{p}} \Vert u \Vert , $$

that is,

$$ \Vert u \Vert _{s}\leq(C_{s})^{-\frac{1}{p}} \Vert u \Vert , $$

where \(\|\cdot\|_{s}\) is the usual norm in \(L^{s}(\varOmega)\).

Definition 2.1

We say that a function \(u\in W^{2,p}(\varOmega)\) is a weak solution of (1.2) if for all \(v\in W^{2,p}(\varOmega)\), we have

$$\begin{aligned} M \biggl( \int_{\varOmega} \bigl( \vert \Delta_{p}u \vert ^{2}+ \vert u \vert ^{p} \bigr)\,dx \biggr) \int_{\varOmega } \bigl( \vert \Delta \vert ^{p-2} \Delta u \Delta v- \vert u \vert ^{p-2}uv \bigr)\,dx =&\lambda \int_{\varOmega}f(x) \vert u \vert ^{q-2}uv\,dx \\ &{}+ \int_{\varOmega}g(x) \vert u \vert ^{m-2}uv\,dx. \end{aligned}$$

Associated with the problem (1.2), we define the functional energy \(J_{\lambda,M}(u):W^{2,p}(\varOmega)\longrightarrow\mathbb{R}\) by

$$ J_{\lambda,M}(u)=\frac{1}{p}\widehat{M} \bigl( \Vert u \Vert ^{p} \bigr)-\frac{\lambda}{q} \int _{\varOmega}f(x) \vert u \vert ^{q}\,dx- \frac{1}{m} \int_{\varOmega}g(x) \vert u \vert ^{m}\,dx, $$

where \(\widehat{M}(t)=at+\frac{b}{l+1}t^{l+1}\).

Lemma 2.1

The functional\(J_{\lambda,M}\)belongs to\(C^{1}(W^{2,p}(\varOmega),\mathbb {R})\). Moreover, for all\(u\in W^{2,p}(\varOmega)\), we have

$$ \bigl\langle J_{\lambda,M}'(u),u\bigr\rangle =a \Vert u \Vert ^{p}+b \Vert u \Vert ^{p(l+1)}-\lambda \int_{\varOmega }f(x) \vert u \vert ^{q}\,dx- \int_{\varOmega}g(x) \vert u \vert ^{m}\,dx, $$

where\(\langle\cdot,\cdot\rangle\)denotes the usual duality between the space\(W^{2,p}(\varOmega)\)and its dual\(W^{-2,p}(\varOmega)\).


From the hypotheses \((H_{1})\)\((H_{2})\) it is obvious that \(J_{\lambda ,M}\in C^{1}(W^{2,p}(\varOmega),\mathbb{R})\) and its Gateaux derivative is given by

$$\begin{aligned} \bigl\langle J_{\lambda,M}'(u),\varphi\bigr\rangle =&M \biggl( \int_{\varOmega} \bigl( \vert \Delta _{p}u \vert ^{2}+ \vert u \vert ^{p} \bigr)\,dx \biggr) \int_{\varOmega} \bigl( \vert \Delta \vert ^{p-2} \Delta u \Delta\varphi- \vert u \vert ^{p-2}u\varphi \bigr)\,dx \\ &{}-\lambda \int_{\varOmega}f(x) \vert u \vert ^{q-2}u\varphi\,dx- \int _{\varOmega}g(x) \vert u \vert ^{m-2}u\varphi\,dx \quad \forall u, \varphi\in W^{2,p}(\varOmega). \end{aligned}$$

This completes the proof of Lemma 2.1. □

Since the energy functional is not bounded from bellow on \(W^{2,p}(\varOmega)\), we introduce the following subspace of \(W^{2,p}(\varOmega)\), which is called Nehari manifold:

$$ N_{\lambda,M}= \bigl\{ u\in W^{2,p}(\varOmega)\setminus\{0\}; \bigl\langle J_{\lambda ,M}'(u),u\bigr\rangle =0 \bigr\} . $$

Thus \(u\in N_{\lambda,M}\) if and only if

$$ a \Vert u \Vert ^{p}+b \Vert u \Vert ^{p(l+1)}-\lambda \int_{\varOmega}f(x) \vert u \vert ^{q}\,dx- \int _{\varOmega}g(x) \vert u \vert ^{m}\,dx=0. $$

Note that the Nehari manifold \(N_{\lambda,M}\) contains every nonzero solution of equation (1.2).

Lemma 2.2

Suppose that\((H_{1})\)and\((H_{2})\)hold. Then the energy functional\(J_{\lambda,M}\)is coercive and bounded below on\(N_{\lambda,M}\).


Let \(u\in N_{\lambda,M}\). Then from (2.1), (2.4), and the Hölder inequality we have

$$\begin{aligned} J_{\lambda,M}(u) =&\frac{1}{p}\widehat{M} \bigl( \Vert u \Vert ^{p} \bigr)-\frac{\lambda }{q} \int_{\varOmega}f(x) \vert u \vert ^{q}\,dx- \frac{1}{m} \int_{\varOmega}g(x) \vert u \vert ^{m}\,dx \\ \geq& \frac{q-p}{pq}a \Vert u \Vert ^{p}+b \biggl( \frac{q-p(l+1)}{qp(l+1)} \biggr) \Vert u \Vert ^{p(l+1)}- \frac{q-m}{mq} \int_{\varOmega}g(x) \vert u \vert ^{m}\,dx \\ \geq& \frac{q-p}{pq}a \Vert u \Vert ^{p}+b \biggl( \frac{q-p(l+1)}{qp(l+1)} \biggr) \Vert u \Vert ^{p(l+1)} \\ &{}-\frac{q-m}{mq} \biggl( \int_{\varOmega} \vert g \vert ^{\frac{p^{*}}{p^{*}-m}}\, dx \biggr)^{\frac{p^{*}-m}{p^{*}}} \biggl( \int_{\varOmega} \vert u \vert ^{p^{*}}\, dx \biggr)^{\frac{m}{p^{*}}} \\ \geq&\frac{q-p}{pq}a \Vert u \Vert ^{p}+b \biggl( \frac{q-p(l+1)}{qp(l+1)} \biggr) \Vert u \Vert ^{p(l+1)}- \frac{q-m}{mq} \Vert g \Vert _{\frac{p^{*}}{p^{*}-m}} (C_{p^{*}} )^{-\frac{m}{p}} \Vert u \Vert ^{m}. \end{aligned}$$

Since \(m< p(l+1)< q\), \(J_{\lambda,M}\) is coercive and bounded below on \(N_{\lambda,M}\). □

The Nehari manifold \(N_{\lambda,M}\) is closely linked to the behavior of the function \(h_{u}:t\longrightarrow J_{\lambda,M}(tu)\) for \(t>0\), defined as follows:

$$ h_{u}(t)=\frac{1}{p}\widehat{M} \bigl(t^{p} \Vert u \Vert ^{p} \bigr)-\lambda\frac {t^{q}}{q} \int_{\varOmega}f(x) \vert u \vert ^{q}\,dx- \frac{t^{m}}{m} \int_{\varOmega }g(x) \vert u \vert ^{m}\,dx. $$

Such maps, introduced by Drábek and Pohozaev [10], are known as fibering maps. A simple calculation shows that, for each \(u\in W^{2,p}(\varOmega)\), we have

$$ h'_{u}(t)=at^{p-1} \Vert u \Vert ^{p}+bt^{p(l+1)-1} \Vert u \Vert ^{p(l+1)}-\lambda t^{q-1} \int_{\varOmega}f(x) \vert u \vert ^{q} \,dx-t^{m-1} \int_{\varOmega}g(x) \vert u \vert ^{m}\,dx $$


$$\begin{aligned} h''_{u}(t) =&a(p-1)t^{p-2} \Vert u \Vert ^{p}+b \bigl(p(l+1)-1 \bigr)t^{p(l+1)-2} \Vert u \Vert ^{p(l+1)} \\ &{}-\lambda(q-1) t^{q-2} \int_{\varOmega}g(x) \vert u \vert ^{q} \,dx-(m-1)t^{m-2} \int _{\varOmega}g(x) \vert u \vert ^{m}\,dx. \end{aligned}$$


$$th'_{u}(t)=\bigl\langle J_{\lambda,M}'(tu),tu\bigr\rangle =0. $$

Thus, for all \(u\in W^{2,p}(\varOmega)\setminus\{0\}\) and \(t>0\), we have

$$h'_{u}(t)=0 \quad\mbox{if and only if}\quad tu\in N_{\lambda,M}. $$

In particular, \(h'_{u}(1)=0\) if and only if \(u\in N_{\lambda,M}\). Also, by equation (2.4) it is easy to see that for \(u\in N_{\lambda,M}\),

$$\begin{aligned} h''_{u}(1) =&a(p-1) \Vert u \Vert ^{p}+b \bigl(p(l+1)-1 \bigr) \Vert u \Vert ^{p(l+1)} \\ &{}-\lambda(q-1) \int_{\varOmega}g(x) \vert u \vert ^{q}\,dx-(m-1) \int_{\varOmega }g(x) \vert u \vert ^{m}\,dx \\ =&a(p-m) \Vert u \Vert ^{p}+b \bigl(p(l+1)-m \bigr) \Vert u \Vert ^{p(l+1)}-\lambda(q-m) \int _{\varOmega}f(x) \vert u \vert ^{q} \,dx \end{aligned}$$
$$\begin{aligned} =& a(p-q) \Vert u \Vert ^{p}+b \bigl(p(l+1)-q \bigr) \Vert u \Vert ^{p(l+1)}+(q-m) \int _{\varOmega}g(x) \vert u \vert ^{m} \,dx \\ =&bpl \Vert u \Vert ^{p(l+1)}+\lambda(p-q) \int_{\varOmega}f(x) \vert u \vert ^{q}\, dx+(p-m) \int_{\varOmega}g(x) \vert u \vert ^{m}\,dx. \end{aligned}$$

In order to have multiplicity of solutions, we split \(N_{\lambda,M}\) into three parts

$$\begin{gathered} N_{\lambda,M}^{+}= \bigl\{ u\in N_{\lambda,M}; h''_{u}(1)>0 \bigr\} , \\ N_{\lambda,M}^{0}= \bigl\{ u\in N_{\lambda,M}; h''_{u}(1)=0 \bigr\} , \\ N_{\lambda,M}^{-}= \bigl\{ u\in N_{\lambda,M}; h''_{u}(1)< 0 \bigr\} .\end{gathered} $$

Furthermore, using arguments similar to those in of Theorem 2.3 in [6], we have the following lemma.

Lemma 2.3

Letube a local minimizer for\(J_{\lambda,M}\)on\(N_{\lambda,M}\)not belonging to\(N_{\lambda,M}^{0}\). Then\(J'_{\lambda,M}(u)=0\).


$$ \lambda_{1}=\frac{a(p-m) (C_{p^{*}} )^{\frac{q}{p}}}{(q-m) \Vert f \Vert _{\frac{p^{*}}{p^{*}-q}}} \biggl(\frac{a(q-p) (C_{p^{*}} )^{\frac {q}{p}}}{(q-m) \Vert g \Vert _{\frac{p^{*}}{p^{*}-m}}} \biggr)^{\frac{q-p}{p-m}}. $$

Then we have the following lemma.

Lemma 2.4

If\(0<|\lambda|<\lambda_{1}\), then\(N_{\lambda ,M}^{0}=\phi\).


Suppose, otherwise, that \(0<|\lambda|<\lambda_{1}\) with \(N_{\lambda ,M}^{0}\neq\phi\). Let \(u\in N_{\lambda,M}^{0}\). Then we have

$$ h_{u}''(1)=0. $$

From (2.5) and (2.6) we get

$$ (q-m) \int_{\varOmega}g(x) \vert u \vert ^{m}\,dx=a(q-p) \Vert u \Vert ^{p}+b \bigl(q-p(l+1) \bigr) \Vert u \Vert ^{p(l+1)} $$


$$ \lambda(q-m) \int_{\varOmega}f(x) \vert u \vert ^{q}\,dx=a(p-m) \Vert u \Vert ^{p}+b \bigl(p(l+1)-m \bigr) \Vert u \Vert ^{p(l+1)}. $$


$$ a(q-p) \Vert u \Vert ^{p}\leq(q-m) \int_{\varOmega}g(x) \vert u \vert ^{m}\,dx $$


$$ a(p-m) \Vert u \Vert ^{p}\leq\lambda(q-m) \int_{\varOmega}f(x) \vert u \vert ^{q}\,dx. $$

On the other hand, from (2.1) and the Hölder inequality we obtain

$$\begin{aligned} (q-m) \int_{\varOmega}g(x) \vert u \vert ^{m}\,dx \leq& (q-m) \Vert g \Vert _{\frac {p^{*}}{p^{*}-m}} (C_{p^{*}} )^{-\frac{m}{p}} \Vert u \Vert ^{m} \end{aligned}$$


$$\begin{aligned} \lambda(q-m) \int_{\varOmega}f(x) \vert u \vert ^{q}\,dx \leq& \vert \lambda \vert (q-m) \Vert f \Vert _{\frac{p^{*}}{p^{*}-q}} (C_{p^{*}} )^{-\frac{q}{p}} \Vert u \Vert ^{q}. \end{aligned}$$

By combining (2.7) and (2.9) we get

$$ \Vert u \Vert \leq \biggl(\frac{(q-m) \Vert g \Vert _{\frac{p^{*}}{p^{*}-m}} (C_{p^{*}} )^{-\frac{m}{p}}}{ a(q-p)} \biggr)^{\frac{1}{p-m}}. $$

Moreover, by combining (2.8) and (2.10) we get

$$ \Vert u \Vert \geq \biggl(\frac{a(p-m) (C_{p^{*}} )^{\frac {q}{p}}}{ \vert \lambda \vert (q-m) \Vert f \Vert _{\frac{p^{*}}{p^{*}-q}}} \biggr)^{\frac{1}{q-p}}. $$

Finally, by combining (2.11) and (2.12) we obtain \(\lambda _{1}\leq|\lambda|\), which is a contradiction. □

From Lemma 2.4, for \(0<|\lambda|<\lambda_{1}\), we can write \(N_{\lambda,M}=N_{\lambda,M}^{+}\cup N_{\lambda,M}^{-}\).


$$ \theta_{\lambda,M}=\inf_{u\in N_{\lambda,M}}J_{\lambda,M}(u),\quad\quad \theta_{\lambda,M}^{+}=\inf_{u\in N_{\lambda,M}^{+}}J_{\lambda ,M}(u)\quad\text{and}\quad \theta_{\lambda,M}^{-}=\inf_{u\in N_{\lambda ,M}^{-}}J_{\lambda,M}(u), $$


$$\lambda_{2}:=\frac{a(p-m) (C_{p^{*}} )^{\frac{q}{p}}}{(q-m) \Vert f \Vert _{\frac{p^{*}}{p^{*}-q}}} \biggl(\frac{ma(q-p) (C_{p^{*}} )^{\frac{m}{p}}}{p(q-m) \Vert g \Vert _{\frac{p^{*}}{p^{*}-m}}} \biggr) ^{\frac{q-p}{p-m}}. $$

Then we have the following:

Proposition 2.1

If\(0< |\lambda|<\lambda_{2}\), then:

  1. (i)
    $$ \theta_{\lambda,M}\leq\theta_{\lambda,M}^{+}< 0. $$
  2. (ii)

    There exists\(C>0\)such that

    $$ \theta_{\lambda,M}^{-}\geq C>0. $$


(i) Let \(u\in N_{\lambda,M}^{+}\). Then from (2.6) and the fact that \(h''_{u}(1)>0\) we obtain

$$ a(q-p) \Vert u \Vert ^{p}+b \bigl(q-p(l+1) \bigr) \Vert u \Vert ^{p(l+1)}< (q-m) \int_{\varOmega }g(x) \vert u \vert ^{m}\,dx. $$

So, by (2.4) we obtain

$$\begin{aligned} J_{\lambda,M}(u) =& a \biggl(\frac{q-p}{pq} \biggr) \Vert u \Vert ^{p}+b \biggl(\frac {q-p(l+1)}{qp(l+1)} \biggr) \Vert u \Vert ^{p(l+1)}-\frac{q-m}{mq} \int_{\varOmega }g(x) \vert u \vert ^{m}\,dx \\ < & \frac{a(q-p)}{q} \biggl(\frac{m-p}{pm} \biggr) \Vert u \Vert ^{p}+\frac {b(q-p(l+1))}{q}\frac{m-p}{m-q} \biggl( \frac{m-p(l+1)}{mp(l+1)} \biggr) \Vert u \Vert ^{p(l+1)}< 0. \end{aligned}$$

Thus we can deduce that \(\theta_{\lambda,M}\leq\theta_{\lambda,M}^{+}<0\).

(ii) Let \(u\in N_{\lambda,M}^{-}\). Then from equations (2.5) and the fact that \(h_{u}''(1)<0\) we get

$$a(p-m) \Vert u \Vert ^{p}+b \bigl(p(l+1)-m \bigr) \Vert u \Vert ^{p(l+1)} < \lambda(q-m) \int_{\varOmega}f(x) \vert u \vert ^{q}. $$


$$a(p-m) \Vert u \Vert ^{p} < \lambda(q-m) \int_{\varOmega}f(x) \vert u \vert ^{q}. $$

Therefore equation (2.10) implies that

$$\Vert u \Vert \geq \biggl(\frac{a(p-m) (C_{p^{*}} )^{\frac {q}{p}}}{ \vert \lambda \vert (q-m) \Vert f \Vert _{\frac{p^{*}}{p^{*}-q}}} \biggr)^{\frac{1}{q-p}}. $$

In addition, from equations (2.1), (2.4), and (2.9), using the Hölder inequality, we have

$$\begin{aligned} J_{\lambda,M}(u) =& a \biggl(\frac{q-p}{pq} \biggr) \Vert u \Vert ^{p}+b \biggl(\frac {q-p(l+1)}{qp(l+1)} \biggr) \Vert u \Vert ^{p(l+1)}-\frac{q-m}{mq} \int_{\varOmega }g(x) \vert u \vert ^{m}\,dx \\ \geq& a \biggl(\frac{q-p}{pq} \biggr) \Vert u \Vert ^{p}-\frac{q-m}{mq} \int _{\varOmega}g(x) \vert u \vert ^{m}\,dx \\ \geq& a \biggl(\frac{q-p}{pq} \biggr) \Vert u \Vert ^{p}-\frac{q-m}{mq} \Vert g \Vert _{\frac{p^{*}}{p^{*}-m}} (C_{p^{*}} )^{-\frac{m}{p}} \Vert u \Vert ^{m} \\ \geq& \Vert u \Vert ^{m} \biggl(a \biggl(\frac{q-p}{pq} \biggr) \Vert u \Vert ^{p-m}-\frac {q-m}{mq} \Vert g \Vert _{\frac{p^{*}}{p^{*}-m}} (C_{p^{*}} )^{-\frac {m}{p}} \biggr) \\ \geq& \biggl(\frac{a(p-m) (C_{p^{*}} )^{\frac {q}{p}}}{ \vert \lambda \vert (q-m) \Vert f \Vert _{\frac{p^{*}}{p^{*}-q}}} \biggr)^{\frac {m}{q-p}} \\ &{}\times\biggl(a \biggl( \frac{q-p}{pq} \biggr) \biggl(\frac{a(p-m) (C_{p^{*}} )^{\frac{q}{p}}}{ \vert \lambda \vert (q-m) \Vert f \Vert _{\frac {p^{*}}{p^{*}-q}}} \biggr)^{\frac{p-m}{q-p}}- \frac{q-m}{mq} \Vert g \Vert _{\frac {p^{*}}{p^{*}-m}} (C_{p^{*}} )^{-\frac{m}{p}} \biggr)\\ :=&C. \end{aligned}$$

It is not difficult to see that if \(0< |\lambda|<\lambda_{2}\), then \(C>0\). This completes the proof of Proposition 2.1. □


$$\lambda_{0}=\min(\lambda_{1}, \lambda_{2}). $$

Proposition 2.2

Suppose that\(0<|\lambda|<\lambda _{0}\). Then for each\(u\in W^{2,p}(\varOmega)\)with

$$ \int_{\varOmega}g(x) \vert u \vert ^{m}\,dx>0, $$

there exists\(T>0\)such that:

  1. (i)

    If\(\lambda \int_{\varOmega}f(x)|u|^{q}\,dx\leq0\), then there exists a unique\(t^{+}< T\)such that\(t^{+}u\in N_{\lambda,M}^{+}\)and

    $$ J_{\lambda,M} \bigl(t^{+}u \bigr)=\inf_{0\leq t\leq T}J_{\lambda,M}(tu). $$
  2. (ii)

    If\(\lambda \int_{\varOmega}f(x)|u|^{q}\,dx> 0\), then there are unique\(0< t^{+}< T< t^{-}\)such that\((t^{-}u,t^{+}u)\in N_{\lambda ,M}^{-}\times N_{\lambda,M}^{+}\)and

    $$ J_{\lambda,M} \bigl(t^{-}u \bigr)=\sup_{t\geq0}J_{\lambda,M}(tu);\qquad J_{\lambda ,M} \bigl(t^{+}u \bigr)=\inf_{0\leq t< T}J_{\lambda,M}(tu). $$


Fix \(u\in W^{2,p}(\varOmega)\) with \(\int_{\varOmega}g(x)|u|^{m}\,dx>0\) and define the map \(\varPsi_{u}\) on \((0,\infty)\) by

$$ \varPsi_{u}(t)=at^{p-q} \Vert u \Vert ^{p}+bt^{p(l+1)-q} \Vert u \Vert ^{p(l+1)}-t^{m-q} \int _{\varOmega}g(x) \vert u \vert ^{m}\,dx. $$

A simple calculation shows that

$$h'_{u}(t)=t^{q-1} \biggl( \varPsi_{u}(t)-\lambda \int_{\varOmega}f(x) \vert u \vert ^{q}\, dx \biggr) . $$

Moreover, for \(t>0\), we have

$$ \varPsi_{u}'(t)=t^{m-q-1} \psi_{u}(t), $$


$$\psi_{u}(t)=a(p-q)t^{p-m} \Vert u \Vert ^{p}+b \bigl(p(l+1)-q \bigr)t^{p(l+1)-m} \Vert u \Vert ^{p(l+1)}+(q-m) \int_{\varOmega}g(x) \vert u \vert ^{m}\,dx. $$

Since \(m< p< p(l+1)< q\), then we have

$$\lim_{t\to0}\psi_{u}(t)=(q-m) \int_{\varOmega}g(x) \vert u \vert ^{m}\,dx>0\quad \mbox{and}\quad \lim_{t\to\infty}\psi_{u}(t)=-\infty. $$

Also, \(\psi_{u}\) is decreasing on \((0,\infty)\). So, there is a unique \(T>0\) such that \(\psi_{u}(t)>0\) for \(0< t< T\), \(\psi_{u}(T)=0\), and \(\psi _{u}(t)<0\) for \(t>T\). Therefore \(\varPsi_{u}\) admits a global maximum at T, \(\psi_{u}\) is increasing on \((0,T)\), decreasing on \((T,\infty)\), \(\lim_{t\to0}\varPsi_{u}(t)=-\infty\), and \(\lim_{t\to\infty}\varPsi_{u}(t)=0\).

(i) If \(\lambda \int_{\varOmega}f(x)|u|^{q}\,dx< 0\), then there is a unique \(t^{+}\in(0,T)\) such that

$$\varPsi_{u} \bigl(t^{+} \bigr)=\lambda \int_{\varOmega}f(x) \vert u \vert ^{q}\,dx\quad \mbox{and}\quad \varPsi'_{u} \bigl(t^{+} \bigr)>0. $$

Therefore \(h'_{u}(t^{+})=0\) and \(h''_{u}(t^{+})>0\), that is, \(h_{u}\) has a global maximum at \(t^{+}\), and \(t^{+}u\in N_{\lambda,M}^{+}\).

(ii) Assume that \(\lambda \int_{\varOmega}f(x)|u|^{q}\,dx>0\), and put

$$ T_{0}= \biggl(\frac{(q-m)\int_{\varOmega}g(x) \vert u \vert ^{m}\,dx}{a(q-p) \Vert u \Vert ^{p}} \biggr)^{\frac{1}{p-m}}. $$

Then we have

$$\psi_{u}(T_{0})=b \bigl(p(l+1)-q \bigr)T_{0}^{p(l+1)-m} \Vert u \Vert ^{p(l+1)}< 0=\psi_{u}(T). $$

Since \(\psi_{u}\) is a decreasing function, we get \(T_{0}>T\). Moreover, since \(\varPsi_{u}\) is decreasing on \((T,\infty)\), from (2.9) we have

$$\begin{aligned} \varPsi_{u}(T) \geq&\varPsi_{u}(T_{0}) \\ \geq&a (T_{0} )^{p-q} \Vert u \Vert ^{p}- (T_{0} )^{m-q} \int_{\varOmega }g(x) \vert u \vert ^{m}\,dx \\ \geq&a \biggl(\frac{a(q-p) \Vert u \Vert ^{p}}{(q-m)\int_{\varOmega}g(x) \vert u \vert ^{m}\, dx} \biggr)^{\frac{q-p}{p-m}} \Vert u \Vert ^{p} \\ &{}- \biggl(\frac{a(q-p) \Vert u \Vert ^{p}}{(q-m)\int_{\varOmega}g(x) \vert u \vert ^{m}\,dx} \biggr)^{\frac{q-m}{p-m}} \int_{\varOmega}g(x) \vert u \vert ^{m}\,dx \\ \geq& \frac{a(p-m)}{q-m} \biggl( \frac{a(q-p)}{q-m} \biggr) ^{ \frac {q-p}{q-m}}\frac{ \Vert u \Vert ^{p\frac{q-m}{p-m}}}{ ( \int_{\varOmega }g(x) \vert u \vert ^{m}\,dx )^{\frac{q-p}{p-m}}} \\ \geq& \frac{a(p-m)}{q-m} \biggl( \frac{a(q-p)}{q-m} \biggr) ^{ \frac {q-p}{q-m}}\frac{ \Vert u \Vert ^{p\frac{q-m}{p-m}}}{ ( (C_{p^{*}} )^{-\frac{m}{p}} \Vert g \Vert _{\frac{p^{*}}{p^{*}-m}} \Vert u \Vert ^{m} )^{\frac {q-p}{p-m}}} \\ \geq& \frac{a(p-m)}{q-m} \biggl( \frac{a(q-p)}{(q-m) (C_{p^{*}} )^{-\frac{m}{p}} \Vert g \Vert _{\frac{p^{*}}{p^{*}-m}}} \biggr) ^{ \frac {q-p}{q-m}} \Vert u \Vert ^{q}. \end{aligned}$$

Therefore by (2.1) we obtain

$$\begin{aligned} \begin{aligned} \varPsi_{u}(T)-\lambda \int_{\varOmega}f(x) \vert u \vert ^{q}\,dx&\geq \frac {a(p-m)}{q-m} \biggl( \frac{a(q-p)}{(q-m) (C_{p^{*}} )^{-\frac {m}{p}} \Vert g \Vert _{\frac{p^{*}}{p^{*}-m}}} \biggr) ^{ \frac{q-p}{q-m}} \Vert u \Vert ^{q}\\&\quad- \vert \lambda \vert \Vert f \Vert _{\frac{p^{*}}{p^{*}-q}}(C_{p^{*}})^{-\frac{q}{p}} \Vert u \Vert ^{q} \\ &\leq \Vert f \Vert _{\frac{p^{*}}{p^{*}-q}}(C_{p^{*}})^{-\frac{q}{p}} \Vert u \Vert ^{q} \bigl(\lambda_{1}- \vert \lambda \vert \bigr).\end{aligned} \end{aligned}$$

Since \(0<|\lambda|<\lambda_{0}\), we have

$$0< \lambda \int_{\varOmega}f(x) \vert u \vert ^{q}\,dx< \varPsi_{u}(T). $$

Hence there are unique \(t^{-}\) and \(t^{+}\) such that \(0< t^{+}< T< t^{-}\),

$$ \varPsi_{u} \bigl(t^{+} \bigr)=\lambda \int_{\varOmega}f(x) \vert u \vert ^{q}\,dx= \varPsi_{u} \bigl(t^{-} \bigr) $$


$$ \varPsi_{u}' \bigl(t^{+} \bigr)>0> \varPsi_{u}' \bigl(t^{-} \bigr). $$

By a similar argument as in case (i) we conclude that \(t^{-}u\in N_{\lambda,M}^{-}\) and \(t^{+}u\in N_{\lambda,M}^{+}\). Moreover,

$$ J_{\lambda,M} \bigl(t^{+}u \bigr)\leq J_{\lambda,M}(tu) \leq J_{\lambda,M} \bigl(t^{-}u \bigr) \quad\text{for each } t \in \bigl[t^{+},t^{-} \bigr], $$

and \(J_{\lambda,M}(tu)\leq J_{\lambda,M}(t^{-}u)\) for each \(t\geq0\). Thus

$$ J_{\lambda,M} \bigl(t^{+}u \bigr)=\inf_{0\leq t\leq T}J_{\lambda,M}(tu)\quad\text{and}\quad J_{\lambda,M} \bigl(t^{-}u \bigr)=\sup_{T\leq t}J_{\lambda,M}(tu). $$


Proposition 2.3

For every\(u\in W^{2,p}(\varOmega)\)with\(\lambda \int_{\varOmega }f(x)|u|^{m}\,dx>0\), there existssuch that:

  1. (i)

    If\(\int_{\varOmega}g(x)|u|^{m}\,dx\leq0\), then there exists a unique\(t^{-}>\widetilde{T}\)such that\(t^{-}u\in N_{\lambda,M}^{-}\)and

    $$ J_{\lambda,M} \bigl(t^{-}u \bigr)=\sup_{t\geq\widetilde{T}}J_{\lambda,M}(tu). $$
  2. (ii)

    If\(\int_{\varOmega}g(x)|u|^{m}\,dx> 0\), then there are unique\(0< t^{+}<\widetilde{T}<t^{-}\)such that\((t^{-}u,t^{+}u)\in N_{\lambda ,M}^{-}\times N_{\lambda,M}^{+}\)and

    $$ J_{\lambda,M} \bigl(t^{-}u \bigr)=\sup_{t\geq0}J_{\lambda,M}(tu);\qquad J_{\lambda ,M} \bigl(t^{+}u \bigr)=\inf_{0\leq t< \widetilde{T}}J_{\lambda,M}(tu). $$


Let \(u\in W^{2,p}(\varOmega)\) be such that \(\lambda\int_{\varOmega }f(x)|u|^{q}\,dx>0\) and define the map \(\varPsi_{u}\) by

$$ \varPsi_{u}(t)=at^{p-m} \Vert u \Vert ^{p}+bt^{p(l+1)-m} \Vert u \Vert ^{p(l+1)}-\lambda t^{q-m} \int_{\varOmega}f(x) \vert u \vert ^{q} \,dx,\quad\text{for } t\geq0. $$


$$ \widetilde{T}_{0}= \biggl(\frac{b (p(l+1)-m ) \Vert u \Vert ^{p(l+1)}}{\lambda (q-m)\int_{\varOmega}f(x) \vert u \vert ^{q}\,dx} \biggr)^{\frac{1}{q-p(l+1)}}. $$

Then by similar arguments as in the proof of Proposition 2.2 we can deduce the results of Proposition 2.3 □

Proposition 2.4

There exist sequences \(\{u_{k}^{\pm}\}\) in \(N_{\lambda}^{\pm}\) such that

$$J_{\lambda,M} \bigl(u_{k}^{\pm} \bigr)= \theta_{\lambda,M}^{\pm}+o(1) \quad\textit{and}\quad J'_{\lambda,M} \bigl(u_{k}^{\pm} \bigr)=o(1). $$


We omit the proof, which is almost the same as that in Wu ([29], Proposition 9). □

3 Proof of our main result

In this section, we apply the method of Nehari manifold combined with the fibering maps to investigate the multiplicity of nontrivial solutions for problem (1.2). To this aim, we assume that \(|\lambda|\in(0,\lambda_{0})\).

Theorem 3.1

Assume that\((H_{1})\)\((H_{2})\)hold. Then problem (1.2) has a nontrivial solution\(u^{+}_{\lambda,M}\)in\(N_{\lambda,M}^{+}\)such that

$$ J_{\lambda,M} \bigl(u^{+}_{\lambda,M} \bigr)= \theta_{\lambda,M}^{+}. $$


By Proposition 2.4 there exists a sequence \(\{u_{k}^{+}\}\) in \(N_{\lambda,M}^{+}\) such that

$$ J_{\lambda,M} \bigl(u_{k}^{+} \bigr)=\theta_{\lambda,M}^{+}+o(1) \quad\text{and}\quad J'_{\lambda,M} \bigl(u_{k}^{+} \bigr)=o(1) \text{ in }W^{-2,p}. $$

Using Lemma 2.2, up to a subsequence, there exists \(u^{+}_{\lambda ,M}\) in \(W^{2,p}(\varOmega)\) such that

$$ \left \{ \textstyle\begin{array}{l} u^{+}_{k}\rightharpoonup u^{+}_{\lambda,M} \quad\text{weakly in } W^{2,p}(\varOmega),\\ u^{+}_{k}\longrightarrow u^{+}_{\lambda,M} \quad\text{strongly in} L^{s}(\varOmega) \text{ for } 1< s< p^{*}, \\ u^{+}_{k}\longrightarrow u^{+}_{\lambda,M} \quad\text{a.e. in } \varOmega. \end{array}\displaystyle \right . $$

We will prove that \(u^{+}_{k}\longrightarrow u^{+}_{\lambda,M}\) strongly in \(W^{2,p}(\varOmega)\) and \(J_{\lambda,M}(u^{+}_{\lambda,M})=\theta^{+}_{\lambda,M}\).

Since \(u^{+}_{\lambda,M}\in N_{\lambda,M}\), by Fatou’s lemma and equation (3.1) we get

$$\begin{aligned} \theta^{+}_{\lambda,M} \leq& J_{\lambda,M} \bigl(u^{+}_{\lambda,M} \bigr)=\frac {1}{p}\widehat{M} \bigl( \bigl\Vert u^{+}_{\lambda,M} \bigr\Vert ^{p} \bigr)- \frac{\lambda}{q} \int_{\varOmega}f(x) \bigl\vert u^{+}_{\lambda,M} \bigr\vert ^{q}\,dx-\frac {1}{m} \int_{\varOmega}g(x) \bigl\vert u^{+}_{\lambda,M} \bigr\vert ^{m}\,dx \\ \leq& \liminf_{k\rightarrow\infty} \biggl(\frac{1}{p}\widehat{M} \bigl( \bigl\Vert u^{+}_{k} \bigr\Vert ^{p} \bigr)- \frac{\lambda}{q} \int_{\varOmega}f(x) \bigl\vert u^{+}_{k} \bigr\vert ^{q}\,dx-\frac {1}{m} \int_{\varOmega}g(x) \bigl\vert u^{+}_{k} \bigr\vert ^{m}\,dx \biggr) \\ \leq& \liminf_{k\rightarrow\infty}J_{\lambda,M} \bigl(u^{+}_{k} \bigr)=\theta _{\lambda,M} \\ =& \lim_{k\uparrow\infty}J_{\lambda,M} \bigl(u^{+}_{k} \bigr)=\theta^{+}_{\lambda,M}. \end{aligned}$$

So, it is easy to see that

$$ J_{\lambda,M} \bigl(u^{+}_{\lambda,M} \bigr)=\theta^{+}_{\lambda,M}\quad\text{and}\quad \widehat{M} \bigl( \bigl\Vert u^{+}_{k} \bigr\Vert ^{p} \bigr)\longrightarrow\widehat{M} \bigl( \bigl\Vert u^{+}_{\lambda ,M} \bigr\Vert ^{p} \bigr) \quad\text{as } k \longrightarrow\infty. $$

From the Brezis–Lieb lemma [5] we obtain \(\| u^{+}_{k}-u^{+}_{\lambda,M}\|^{p}=\|u^{+}_{k}\|^{p}-\|u^{+}_{\lambda,M}\|^{p}\). Therefore \(u_{k}\longrightarrow u_{\lambda,M}\) strongly in \(W^{2,p}(\varOmega)\).

Now we will prove that \(u^{+}_{\lambda,M}\in N_{\lambda,M}^{+}\). We proceed by contradiction assuming that \(u^{+}_{\lambda,M}\in N_{\lambda ,M}^{-}\).

We have

$$\begin{aligned} J_{\lambda,M} \bigl(u^{+}_{k} \bigr) =& \frac{q-p}{pq}a \bigl\Vert u^{+}_{k} \bigr\Vert ^{p}+b \biggl( \frac {q-p(l+1)}{qp(l+1)} \biggr) \bigl\Vert u^{+}_{k} \bigr\Vert ^{p(l+1)}-\frac{q-m}{mq} \int _{\varOmega}g(x) \bigl\vert u^{+}_{k} \bigr\vert ^{m}\,dx \\ \geq&-\frac{q-m}{mq} \int_{\varOmega}g(x) \bigl\vert u^{+}_{k} \bigr\vert ^{m}\,dx. \end{aligned}$$

By letting k tend to infinity we obtain

$$\int_{\varOmega}g(x) \bigl\vert u^{+}_{\lambda,M} \bigr\vert ^{m}\,dx\geq- \frac{mq}{q-m} \theta _{\lambda,M}^{+}>0. $$

Therefore \(u^{+}_{\lambda,M}\) is nontrivial. Moreover, Propositions 2.2 and 2.3(ii) imply the existence of a unique \(t^{+}\) such that \(t^{+}u^{+}_{\lambda,M}\in N_{\lambda,M}^{+}\). Since \(u^{+}_{\lambda,M}\in N_{\lambda,M}^{-}\), we have

$$ \frac{d^{2}}{dt^{2}}h_{u^{+}_{\lambda,M}} \bigl(t^{+} \bigr)>0 \quad\text{and} \quad\frac {d}{dt}h_{u^{+}_{\lambda,M}}(1)< 0. $$

So, there exists \(\widetilde{t}\in(t^{+},1)\) such that

$$h_{u^{+}_{\lambda,M}} \bigl(t^{+} \bigr)=J_{\lambda,M} \bigl(t^{+}u^{+}_{\lambda,M} \bigr)< h_{ u^{+}_{\lambda,M}}( \widetilde{t})=J_{\lambda,M} \bigl(\widetilde {t}u^{+}_{\lambda,M} \bigr). $$


$$ J_{\lambda,M} \bigl(t^{+}u^{+}_{\lambda,M} \bigr)< J_{\lambda,M} \bigl(\widetilde {t}u^{+}_{\lambda,M} \bigr)\leq J_{\lambda,M} \bigl(t^{-}u^{+}_{\lambda ,M} \bigr)=J_{\lambda,M} \bigl(u^{+}_{\lambda,M} \bigr), $$

which is a contradiction. Therefore \(u^{+}_{\lambda,M}\in N_{\lambda ,M}^{+}\), Moreover, it is not difficult to see that (3.1) and (3.2) imply that \(u^{+}_{\lambda,M}\) is a weak solution of problem (1.2). The proof is now completed. □

Theorem 3.2

If\(0<|\lambda|<\lambda_{0}\)and\((H_{1})\)\((H_{3})\)hold, then problem (1.2) admits a nontrivial solution\(u^{-}_{\lambda,M}\)in\(N_{\lambda,M}^{-}\)satisfying

$$ J_{\lambda,M} \bigl(u^{-}_{\lambda,M} \bigr)= \theta_{\lambda,M}^{-}. $$


By Proposition 2.4 there exists a sequence \(\{u^{-}_{k}\}\) in \(N_{\lambda,M}^{-}\) such that

$$ J_{\lambda,M} \bigl(u^{-}_{k} \bigr)=\theta_{\lambda,M}^{-}+o(1) \quad\text{and}\quad J'_{\lambda,M} \bigl(u^{-}_{k} \bigr)=o(1) \quad\text{in }W^{-2,p}(\varOmega). $$

Using Lemma 2.2, up to a subsequence, there exists \(u^{-}_{\lambda ,M}\) in \(W^{2,p}(\varOmega)\) such that

$$ \left \{ \textstyle\begin{array}{l} u^{-}_{k}\rightharpoonup u^{-}_{\lambda,M} \quad\text{weakly in } W^{2,p}(\varOmega),\\ u^{-}_{k}\longrightarrow u^{-}_{\lambda,M} \quad\text{strongly in} L^{s}(\varOmega) \text{ for } 1< s< p^{*}, \\ u^{-}_{k}\longrightarrow u^{-}_{\lambda,M} \quad\text{a.e. in } \varOmega. \end{array}\displaystyle \right . $$

We begin by proving that the sequence \(\{u^{-}_{k}\}\) converges strongly to \(u^{-}_{\lambda,M}\) in \(W^{2,p}(\varOmega)\). Suppose that, on the contrary,

$$ \bigl\Vert u^{-}_{\lambda,M} \bigr\Vert < \liminf _{k\longrightarrow\infty} \bigl\Vert u^{-}_{k} \bigr\Vert . $$

Since \(u^{-}_{k}\in N_{\lambda,M}^{-}\), from equations (2.5) and the fact that \(h_{u^{-}_{k}}''(1)<0\) we get

$$a(p-m) \bigl\Vert u^{-}_{k} \bigr\Vert ^{p}+b \bigl(p(l+1)-m \bigr) \bigl\Vert u^{-}_{k} \bigr\Vert ^{p(l+1)} < \lambda(q-m) \int_{\varOmega}f(x) \bigl\vert u^{-}_{k} \bigr\vert ^{q}, $$

which implies that

$$ a(p-m) \bigl\Vert u^{-}_{k} \bigr\Vert ^{p} < \lambda(q-m) \int_{\varOmega}f(x) \bigl\vert u^{-}_{k} \bigr\vert ^{q}. $$

Therefore equation (2.10) implies that

$$ \bigl\Vert u^{-}_{k} \bigr\Vert \geq \biggl(\frac{a(p-m) (C_{p^{*}} )^{\frac {q}{p}}}{ \vert \lambda \vert (q-m) \Vert f \Vert _{\frac{p^{*}}{p^{*}-q}}} \biggr)^{\frac{1}{q-p}}. $$

By combining (3.5) and (3.6) we obtain

$$\lambda \int_{\varOmega}f(x) \bigl\vert u^{-}_{k} \bigr\vert ^{q}>\frac{a(p-m)}{q-m} \biggl(\frac {a(p-m) (C_{p^{*}} )^{\frac{q}{p}}}{ \vert \lambda \vert (q-m) \Vert f \Vert _{\frac {p^{*}}{p^{*}-q}}} \biggr)^{\frac{p}{q-p}}. $$

Passing to the limits as k tends to infinity, we obtain

$$\lambda \int_{\varOmega}f(x) \bigl\vert u^{-}_{\lambda,M} \bigr\vert ^{q}\geq\frac {a(p-m)}{q-m} \biggl( \frac{a(p-m) (C_{p^{*}} )^{\frac {q}{p}}}{ \vert \lambda \vert (q-m) \Vert f \Vert _{\frac{p^{*}}{p^{*}-q}}} \biggr)^{\frac{p}{q-p}}>0. $$

Therefore \(u^{-}_{\lambda,M}\) is nontrivial. Moreover, by Proposition 2.3 there exist a unique \(t^{-}>0\) such that \(t^{-}u^{-}_{\lambda,M}\in N_{\lambda,M}^{-}\). Therefore

$$ J_{\lambda,M} \bigl(t^{-}u^{-}_{\lambda,M} \bigr)< \lim_{k\longrightarrow\infty }J_{\lambda,M} \bigl(t^{-}u^{-}_{k} \bigr)\leq\lim_{k\longrightarrow\infty }J_{\lambda,M} \bigl(u^{-}_{k} \bigr)=\theta_{\lambda,M}^{-}, $$

a contradiction. Hence \(u^{-}_{k}\longrightarrow u^{-}_{\lambda,M}\) strongly in \(W^{2,p}(\varOmega)\). This implies that

$$ J_{\lambda,M} \bigl(u^{-}_{k} \bigr) \longrightarrow J_{\lambda,M} \bigl(u^{-}_{\lambda ,M} \bigr)= \theta_{\lambda,M}^{-} \quad\text{as } n\longrightarrow \infty. $$

Finally, from (3.3) and (3.4) we obtain that \(u^{-}_{\lambda,M}\) is a weak solution of problem (1.2). This ends the proof of Theorem 3.2 . □

Now Theorems 3.1 and 3.2 and the fact that \(N_{\lambda ,M}^{-}\cap N_{\lambda,M}^{-}=\emptyset\) finishes the proof of Theorem 1.1.


  1. Alsaedi, R.: Perturbed subcritical Dirichlet problems with variable exponents. Electron. J. Differ. Equ. 2016, Article ID 295 (2016)

    Article  MathSciNet  Google Scholar 

  2. Alsaedi, R., Dhifli, A., Ghanmi, A.: Low perturbations of p-biharmonic equations with competing nonlinearities. Complex Var. Elliptic Equ. (2020).

    Article  Google Scholar 

  3. Alsaedi, R., Mâagli, H., Rǎdulescu, V.D., Zeddini, N.: Asymptotic behavior of positive large solutions of quasilinear logistic problems. Electron. J. Qual. Theory Differ. Equ. 2015, Article ID 28 (2015)

    Article  Google Scholar 

  4. Ben Ali, K., Ghanmi, A., Kefi, K.: Minimax method involving singular \(p(x)\)-Kirchhoff equation. J. Math. Phys. 58, Article ID 111505 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  5. Brezis, H., Lieb, E.: A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc. 88, 486–490 (1983)

    Article  MathSciNet  Google Scholar 

  6. Brown, K.J., Zhang, Y.: The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function. J. Differ. Equ. 193, 481–499 (2003)

    Article  MathSciNet  Google Scholar 

  7. Chammem, R., Ghanmi, A., Sahbani, A.: Existence of solution for a singular fractional Laplacian problem with variable exponents and indefinite weights. Complex Var. Elliptic Equ. (2020).

    Article  Google Scholar 

  8. Chen, C.Y., Kuo, Y.-C., Wu, T.F.: The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions. J. Differ. Equ. 250, 1876–1908 (2011)

    Article  MathSciNet  Google Scholar 

  9. Chen, Y., Levine, S., Rao, R.: Variable exponent, linear growth functionals in image processing. SIAM J. Appl. Math. 66, 1383–1406 (2006)

    Article  MathSciNet  Google Scholar 

  10. Dràbek, P., Pohozaev, S.I.: Positive solutions for the p-Laplacian: application of the fibering method. Proc. R. Soc. Edinb., Sect. A 127, 703–726 (1997)

    Article  MathSciNet  Google Scholar 

  11. Ghanmi, A.: Multiplicity of nontrivial solutions of a class of fractional p-Laplacian problem. Z. Anal. Anwend. 34, 309–319 (2015)

    Article  MathSciNet  Google Scholar 

  12. Ghanmi, A.: Existence of nonnegative solutions for a class of fractional p-Laplacian problems. Nonlinear Stud. 22(3), 373–379 (2015)

    MathSciNet  MATH  Google Scholar 

  13. Ghanmi, A., Saoudi, K.: A multiplicity results for a singular problem involving the fractional p-Laplacian operator. Complex Var. Elliptic Equ. 61(9), 1199–1216 (2016)

    Article  MathSciNet  Google Scholar 

  14. Groza, Gh., Ali Khan, S.M., Pop, N.: Approximate solutions of boundary value problems for ODEs using Newton interpolating series. Carpath. J. Math. 25(1), 73–81 (2009)

    MathSciNet  MATH  Google Scholar 

  15. Ji, C.: The Nehari manifold for a degenerate elliptic equation involving a sign-changing weight function. Nonlinear Anal. 75, 806–818 (2012)

    Article  MathSciNet  Google Scholar 

  16. Ji, C., Wang, W.H.: On the p-biharmonic equation involving concave-convex nonlinearities and sign-changing weight function. Electron. J. Qual. Theory Differ. Equ. 2012, Article ID 2 (2012)

    Article  MathSciNet  Google Scholar 

  17. Lazer, A.C., McKenna, P.J.: Large-amplitude periodic oscillations in suspension bridge: some new connections with nonlinear analysis. SIAM Rev. 32, 537–578 (1990)

    Article  MathSciNet  Google Scholar 

  18. Mâagli, H., Alsaedi, R., Zeddini, N.: Bifurcation analysis of elleptic equations described by nonhomogeneous differential operators. Electron. J. Differ. Equ. 2017, Article ID 223 (2017)

    Article  Google Scholar 

  19. Marin, M., Cractun, E.M., Pop, N.: Considerations on mixed initial-boundary value problems for micropolar porous bodies. Dyn. Syst. Appl. 25, 175–196 (2016)

    MATH  Google Scholar 

  20. McKenna, P.J., Walter, W.: Nonlinear oscillation in a suspension bridge. Arch. Ration. Mech. Anal. 98, 167–177 (1987)

    Article  MathSciNet  Google Scholar 

  21. McKenna, P.J., Walter, W.: Traveling waves in a suspension bridge. SIAM J. Appl. Math. 50, 703–715 (1990)

    Article  MathSciNet  Google Scholar 

  22. Molica Bisci, G., Rǎdulescu, V.D.: Ground state solution of scalar field fractional Schrödinger equations. Calc. Var. Partial Differ. Equ. 54(3), 2985–3008 (2015)

    Article  Google Scholar 

  23. Molica Bisci, G., Rǎdulescu, V.D., Servadei, R.: Variational Methods for Nonlocal Fractional Problems. Encyclopedia of Mathematics and Its Applications, vol. 162. Cambridge University Press, Cambridge (2016)

    Book  Google Scholar 

  24. Rǎdulescu, V.D.: Nonlinear elliptic equations with variable exponent: old and new. Nonlinear Anal., Theory Methods Appl. 121, 336–369 (2015)

    Article  MathSciNet  Google Scholar 

  25. Ruzicka, M.: Flow of shear dependent electrorheological uids. C. R. Math. Acad. Sci. Paris 329, 393–398 (1999)

    Article  Google Scholar 

  26. Sun, J., Chu, J., Wu, T.F.: Existence and multiplicity of nontrivial solutions for some biharmonic equations with p-Laplacian. J. Differ. Equ. 262, 945–977 (2017)

    Article  MathSciNet  Google Scholar 

  27. Sun, J., Wu, T.F.: Existence of nontrivial solutions for a biharmonic equation with p-Laplacian and singular sign-changing potential. Appl. Math. Lett. 66, 61–67 (2017)

    Article  MathSciNet  Google Scholar 

  28. Sun, J., Wu, T.F.: The Nehari manifold of biharmonic equations with p-Laplacian and singular potential. Appl. Math. Lett. 88, 156–163 (2019)

    Article  MathSciNet  Google Scholar 

  29. Wu, T.F.: On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function. J. Math. Anal. 318, 253–270 (2006)

    Article  MathSciNet  Google Scholar 

  30. Zhikov, V.: Averaging of functionals in the calculus of variations and elasticity. Math. USSR, Izv. 29, 33–66 (1987)

    Article  Google Scholar 

Download references


The author would like to thank the anonymous referees for valuable suggestions and comments, which improved the quality of this paper.

Availability of data and materials

Not applicable.


Not applicable.

Author information

Authors and Affiliations



The author read and approved the final manuscript.

Corresponding author

Correspondence to Ramzi Alsaedi.

Ethics declarations

Competing interests

The author declares that there is no conflict of interest regarding the publication of this paper.

Additional information


Not applicable.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Alsaedi, R. Multiplicity results involving p-biharmonic Kirchhoff-type problems. Bound Value Probl 2020, 118 (2020).

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI:


  • 31B30
  • 35J35
  • 74H20


  • Variational method
  • Biharmonic Kirchhoff-type equations
  • Multiple solutions
  • Nehari manifold