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,
$$ \Vert u \Vert _{s}\leq(C_{s})^{-\frac{1}{p}} \Vert u \Vert , $$
(2.1)
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
$$ J_{\lambda,M}(u)=\frac{1}{p}\widehat{M} \bigl( \Vert u \Vert ^{p} \bigr)-\frac{\lambda}{q} \int _{\varOmega}f(x) \vert u \vert ^{q}\,dx- \frac{1}{m} \int_{\varOmega}g(x) \vert u \vert ^{m}\,dx, $$
(2.2)
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
$$ \bigl\langle J_{\lambda,M}'(u),u\bigr\rangle =a \Vert u \Vert ^{p}+b \Vert u \Vert ^{p(l+1)}-\lambda \int_{\varOmega }f(x) \vert u \vert ^{q}\,dx- \int_{\varOmega}g(x) \vert u \vert ^{m}\,dx, $$
(2.3)
where\(\langle\cdot,\cdot\rangle\)denotes the usual duality between the space\(W^{2,p}(\varOmega)\)and its dual\(W^{-2,p}(\varOmega)\).
Proof
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
$$ a \Vert u \Vert ^{p}+b \Vert u \Vert ^{p(l+1)}-\lambda \int_{\varOmega}f(x) \vert u \vert ^{q}\,dx- \int _{\varOmega}g(x) \vert u \vert ^{m}\,dx=0. $$
(2.4)
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\(N_{\lambda,M}\).
Proof
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
$$ h'_{u}(t)=at^{p-1} \Vert u \Vert ^{p}+bt^{p(l+1)-1} \Vert u \Vert ^{p(l+1)}-\lambda t^{q-1} \int_{\varOmega}f(x) \vert u \vert ^{q} \,dx-t^{m-1} \int_{\varOmega}g(x) \vert u \vert ^{m}\,dx $$
and
$$\begin{aligned} h''_{u}(t) =&a(p-1)t^{p-2} \Vert u \Vert ^{p}+b \bigl(p(l+1)-1 \bigr)t^{p(l+1)-2} \Vert u \Vert ^{p(l+1)} \\ &{}-\lambda(q-1) t^{q-2} \int_{\varOmega}g(x) \vert u \vert ^{q} \,dx-(m-1)t^{m-2} \int _{\varOmega}g(x) \vert u \vert ^{m}\,dx. \end{aligned}$$
Clearly,
$$th'_{u}(t)=\bigl\langle J_{\lambda,M}'(tu),tu\bigr\rangle =0. $$
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 \(u\in N_{\lambda,M}\),
$$\begin{aligned} h''_{u}(1) =&a(p-1) \Vert u \Vert ^{p}+b \bigl(p(l+1)-1 \bigr) \Vert u \Vert ^{p(l+1)} \\ &{}-\lambda(q-1) \int_{\varOmega}g(x) \vert u \vert ^{q}\,dx-(m-1) \int_{\varOmega }g(x) \vert u \vert ^{m}\,dx \\ =&a(p-m) \Vert u \Vert ^{p}+b \bigl(p(l+1)-m \bigr) \Vert u \Vert ^{p(l+1)}-\lambda(q-m) \int _{\varOmega}f(x) \vert u \vert ^{q} \,dx \end{aligned}$$
(2.5)
$$\begin{aligned} =& a(p-q) \Vert u \Vert ^{p}+b \bigl(p(l+1)-q \bigr) \Vert u \Vert ^{p(l+1)}+(q-m) \int _{\varOmega}g(x) \vert u \vert ^{m} \,dx \\ =&bpl \Vert u \Vert ^{p(l+1)}+\lambda(p-q) \int_{\varOmega}f(x) \vert u \vert ^{q}\, dx+(p-m) \int_{\varOmega}g(x) \vert u \vert ^{m}\,dx. \end{aligned}$$
(2.6)
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\(J'_{\lambda,M}(u)=0\).
Put
$$ \lambda_{1}=\frac{a(p-m) (C_{p^{*}} )^{\frac{q}{p}}}{(q-m) \Vert f \Vert _{\frac{p^{*}}{p^{*}-q}}} \biggl(\frac{a(q-p) (C_{p^{*}} )^{\frac {q}{p}}}{(q-m) \Vert g \Vert _{\frac{p^{*}}{p^{*}-m}}} \biggr)^{\frac{q-p}{p-m}}. $$
Then we have the following lemma.
Lemma 2.4
If\(0<|\lambda|<\lambda_{1}\), then\(N_{\lambda ,M}^{0}=\phi\).
Proof
Suppose, otherwise, that \(0<|\lambda|<\lambda_{1}\) with \(N_{\lambda ,M}^{0}\neq\phi\). Let \(u\in N_{\lambda,M}^{0}\). Then we have
From (2.5) and (2.6) we get
$$ (q-m) \int_{\varOmega}g(x) \vert u \vert ^{m}\,dx=a(q-p) \Vert u \Vert ^{p}+b \bigl(q-p(l+1) \bigr) \Vert u \Vert ^{p(l+1)} $$
and
$$ \lambda(q-m) \int_{\varOmega}f(x) \vert u \vert ^{q}\,dx=a(p-m) \Vert u \Vert ^{p}+b \bigl(p(l+1)-m \bigr) \Vert u \Vert ^{p(l+1)}. $$
Therefore
$$ a(q-p) \Vert u \Vert ^{p}\leq(q-m) \int_{\varOmega}g(x) \vert u \vert ^{m}\,dx $$
(2.7)
and
$$ a(p-m) \Vert u \Vert ^{p}\leq\lambda(q-m) \int_{\varOmega}f(x) \vert u \vert ^{q}\,dx. $$
(2.8)
On the other hand, from (2.1) and the Hölder inequality we obtain
$$\begin{aligned} (q-m) \int_{\varOmega}g(x) \vert u \vert ^{m}\,dx \leq& (q-m) \Vert g \Vert _{\frac {p^{*}}{p^{*}-m}} (C_{p^{*}} )^{-\frac{m}{p}} \Vert u \Vert ^{m} \end{aligned}$$
(2.9)
and
$$\begin{aligned} \lambda(q-m) \int_{\varOmega}f(x) \vert u \vert ^{q}\,dx \leq& \vert \lambda \vert (q-m) \Vert f \Vert _{\frac{p^{*}}{p^{*}-q}} (C_{p^{*}} )^{-\frac{q}{p}} \Vert u \Vert ^{q}. \end{aligned}$$
(2.10)
By combining (2.7) and (2.9) we get
$$ \Vert u \Vert \leq \biggl(\frac{(q-m) \Vert g \Vert _{\frac{p^{*}}{p^{*}-m}} (C_{p^{*}} )^{-\frac{m}{p}}}{ a(q-p)} \biggr)^{\frac{1}{p-m}}. $$
(2.11)
Moreover, by combining (2.8) and (2.10) we get
$$ \Vert u \Vert \geq \biggl(\frac{a(p-m) (C_{p^{*}} )^{\frac {q}{p}}}{ \vert \lambda \vert (q-m) \Vert f \Vert _{\frac{p^{*}}{p^{*}-q}}} \biggr)^{\frac{1}{q-p}}. $$
(2.12)
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 \(N_{\lambda,M}=N_{\lambda,M}^{+}\cup N_{\lambda,M}^{-}\).
Put
$$ \theta_{\lambda,M}=\inf_{u\in N_{\lambda,M}}J_{\lambda,M}(u),\quad\quad \theta_{\lambda,M}^{+}=\inf_{u\in N_{\lambda,M}^{+}}J_{\lambda ,M}(u)\quad\text{and}\quad \theta_{\lambda,M}^{-}=\inf_{u\in N_{\lambda ,M}^{-}}J_{\lambda,M}(u), $$
and
$$\lambda_{2}:=\frac{a(p-m) (C_{p^{*}} )^{\frac{q}{p}}}{(q-m) \Vert f \Vert _{\frac{p^{*}}{p^{*}-q}}} \biggl(\frac{ma(q-p) (C_{p^{*}} )^{\frac{m}{p}}}{p(q-m) \Vert g \Vert _{\frac{p^{*}}{p^{*}-m}}} \biggr) ^{\frac{q-p}{p-m}}. $$
Then we have the following:
Proposition 2.1
If\(0< |\lambda|<\lambda_{2}\), then:
-
(i)
$$ \theta_{\lambda,M}\leq\theta_{\lambda,M}^{+}< 0. $$
(2.13)
-
(ii)
There exists\(C>0\)such that
$$ \theta_{\lambda,M}^{-}\geq C>0. $$
(2.14)
Proof
(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
$$a(p-m) \Vert u \Vert ^{p}+b \bigl(p(l+1)-m \bigr) \Vert u \Vert ^{p(l+1)} < \lambda(q-m) \int_{\varOmega}f(x) \vert u \vert ^{q}. $$
So,
$$a(p-m) \Vert u \Vert ^{p} < \lambda(q-m) \int_{\varOmega}f(x) \vert u \vert ^{q}. $$
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. □
Set
$$\lambda_{0}=\min(\lambda_{1}, \lambda_{2}). $$
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:
-
(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
$$ J_{\lambda,M} \bigl(t^{+}u \bigr)=\inf_{0\leq t\leq T}J_{\lambda,M}(tu). $$
-
(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
$$ J_{\lambda,M} \bigl(t^{-}u \bigr)=\sup_{t\geq0}J_{\lambda,M}(tu);\qquad J_{\lambda ,M} \bigl(t^{+}u \bigr)=\inf_{0\leq t< T}J_{\lambda,M}(tu). $$
Proof
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
$$ \varPsi_{u}'(t)=t^{m-q-1} \psi_{u}(t), $$
where
$$\psi_{u}(t)=a(p-q)t^{p-m} \Vert u \Vert ^{p}+b \bigl(p(l+1)-q \bigr)t^{p(l+1)-m} \Vert u \Vert ^{p(l+1)}+(q-m) \int_{\varOmega}g(x) \vert u \vert ^{m}\,dx. $$
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 \(0< t^{+}< T< t^{-}\),
$$ \varPsi_{u} \bigl(t^{+} \bigr)=\lambda \int_{\varOmega}f(x) \vert u \vert ^{q}\,dx= \varPsi_{u} \bigl(t^{-} \bigr) $$
and
$$ \varPsi_{u}' \bigl(t^{+} \bigr)>0> \varPsi_{u}' \bigl(t^{-} \bigr). $$
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 existsT̃such that:
-
(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
$$ J_{\lambda,M} \bigl(t^{-}u \bigr)=\sup_{t\geq\widetilde{T}}J_{\lambda,M}(tu). $$
-
(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
$$ J_{\lambda,M} \bigl(t^{-}u \bigr)=\sup_{t\geq0}J_{\lambda,M}(tu);\qquad J_{\lambda ,M} \bigl(t^{+}u \bigr)=\inf_{0\leq t< \widetilde{T}}J_{\lambda,M}(tu). $$
Proof
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
$$ \varPsi_{u}(t)=at^{p-m} \Vert u \Vert ^{p}+bt^{p(l+1)-m} \Vert u \Vert ^{p(l+1)}-\lambda t^{q-m} \int_{\varOmega}f(x) \vert u \vert ^{q} \,dx,\quad\text{for } t\geq0. $$
Put
$$ \widetilde{T}_{0}= \biggl(\frac{b (p(l+1)-m ) \Vert u \Vert ^{p(l+1)}}{\lambda (q-m)\int_{\varOmega}f(x) \vert u \vert ^{q}\,dx} \biggr)^{\frac{1}{q-p(l+1)}}. $$
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
$$J_{\lambda,M} \bigl(u_{k}^{\pm} \bigr)= \theta_{\lambda,M}^{\pm}+o(1) \quad\textit{and}\quad J'_{\lambda,M} \bigl(u_{k}^{\pm} \bigr)=o(1). $$
Proof
We omit the proof, which is almost the same as that in Wu ([29], Proposition 9). □