In this section, we will consider the existence of a positive periodic solution for (1.5) with singularity.
Theorem 3.1
Assume that the following conditions hold:
-
\((H_{1})\)
:
-
there exists a positive constant
K
such that
\(\vert f(t,v)\vert \leq K\), for
\((t,v)\in \mathbb{R}\times \mathbb{R}\);
-
\((H_{2})\)
:
-
there exist positive constants
\(D_{1}\)
and
\(D_{2}\)
with
\(D_{1}>D_{2}>0\)
such that
\(g(t,u)<-K\)
for
\((t,u)\in \mathbb{R}\times (D_{1},+\infty)\)
and
\(g(t,u)>K\)
for
\((t,u)\in \mathbb{R}\times (0,D _{2})\);
-
\((H_{3})\)
:
-
there exist positive constants
a, b
such that
$$-g(t,u)\leq a u^{p-1}+b,\quad \textit{for all } u>0. $$
Then (1.5) has at least one positive solution with period
ω
if
\(\frac{\omega (1+\vert c\vert )^{\frac{1}{p}}a^{\frac{1}{p}}}{\vert 1-\vert c\vert \vert }<2^{ \frac{p-1}{p}}\).
Proof
Firstly, we will claim that the set of all possible ω-periodic solutions of (2.1) is bounded. Let \(u(t)\in C^{1}_{\omega }\) be an arbitrary solution of (2.1) with period ω.
We claim that there exists a point \(t_{0}\in [0,\omega ]\) such that
$$ 0< u(t_{0})\leq D_{1}. $$
(3.1)
Integrating both sides of (2.1) over \([0,\omega ]\), we have
$$ \int^{\omega }_{0} \bigl[f \bigl(t,u'(t) \bigr)+g \bigl(t,u(t) \bigr) \bigr]\,dt=0. $$
(3.2)
Therefore, from \((H_{1})\), we have
$$-K\omega \leq \int^{\omega }_{0}g \bigl(t,u(t) \bigr)\,dt\leq K\omega. $$
From \((H_{2})\), we know that there exist two points \(t_{0}\), \(\tau \in (0,T)\), such that
$$ u(t_{0})\leq D_{1}, \quad \mbox{and}\quad u( \tau)>D_{2}. $$
(3.3)
Since \(u(t)>0\), \(t\in [0,\omega ]\), we get \(0< u(t_{0})\leq D_{1}\). Equation (3.1) is proved.
Then, from Lemma 2.3, we have
$$ \begin{aligned} \Vert u\Vert _{\infty } & \leq D_{1}+\frac{1}{2} \int^{\omega }_{0} \bigl\vert u'(s) \bigr\vert \,ds. \end{aligned} $$
(3.4)
Multiplying both sides of (2.1) by \((Au)(t)\) and integrating over \([0,\omega ]\), we get
$$\begin{aligned}& \int^{\omega }_{0} \bigl(\phi_{p}(Au)'(t) \bigr)'(Au) (t)\,dt+\lambda \int^{\omega } _{0}f \bigl(t,u'(t) \bigr) (Au) (t)\,dt+\lambda \int^{\omega }_{0}g \bigl(t,u(t) \bigr) (Au) (t)\,dt \\& \quad = \lambda \int^{\omega }_{0} e(t) (Au) (t)\,dt, \end{aligned}$$
i.e.
$$\begin{aligned} \int^{\omega }_{0} \bigl\vert (Au)'(t) \bigr\vert ^{p}\,dt =&\lambda \int^{\omega }_{0}f \bigl(t,u'(t) \bigr) (Au) (t)\,dt+ \lambda \int^{\omega }_{0}g \bigl(t,u(t) \bigr) (Au) (t)\,dt \\ &{}- \lambda \int^{\omega } _{0} e(t) (Au) (t)\,dt. \end{aligned}$$
(3.5)
From \((H_{1})\), we have
$$\begin{aligned}& \int^{\omega }_{0} \bigl\vert (Au)'(t) \bigr\vert ^{p}\,dt \\& \quad \leq \bigl(1+\vert c\vert \bigr) \int^{\omega }_{0} \bigl\vert f \bigl(t,u'(t) \bigr) \bigr\vert \bigl\vert u(t) \bigr\vert \,dt+ \int^{\omega }_{0} \bigl\vert g \bigl(t,u(t) \bigr) \bigr\vert \bigl\vert u(t) \bigr\vert \,dt+ \int^{\omega }_{0} \bigl\vert e(t) \bigr\vert \bigl\vert u(t) \bigr\vert \,dt \\& \quad \leq \bigl(1+\vert c\vert \bigr)\Vert u\Vert _{\infty } \biggl( \int^{\omega }_{0} \bigl\vert f \bigl(t,u'(t) \bigr) \bigr\vert \,dt+ \int^{\omega }_{0} \bigl\vert g \bigl(t,u(t) \bigr) \bigr\vert \,dt + \int^{\omega }_{0} \bigl\vert e(t) \bigr\vert \,dt \biggr) \\& \quad \leq \bigl(1+\vert c\vert \bigr)\Vert u\Vert _{\infty } \biggl( K \omega +\Vert e\Vert _{\infty }\omega + \int^{\omega }_{0} \bigl\vert g \bigl(t,u(t) \bigr) \bigr\vert \,dt \biggr). \end{aligned}$$
(3.6)
We get from \((H_{1})\), \((H_{3})\) and (3.2)
$$\begin{aligned} \int^{\omega }_{0} \bigl\vert g \bigl(t,u(t) \bigr) \bigr\vert \,dt = & \int_{g(t,u(t))\geq 0}g^{+} \bigl(t,u(t) \bigr)\,dt- \int_{g(t,u(t))\leq 0}g^{-} \bigl(t,u(t) \bigr)\,dt \\ = &-2 \int_{g(t,u(t))\leq 0 }g^{-} \bigl(t,u(t) \bigr)\,dt+ \int^{\omega }_{0}f \bigl(t,u'(t) \bigr)\,dt \\ \leq &2a \int^{\omega }_{0} \bigl\vert u(t) \bigr\vert ^{p-1}\,dt+2b\omega +K\omega \\ \leq &2a\omega \Vert u\Vert _{\infty }^{p-1}+2b\omega +K \omega, \end{aligned}$$
(3.7)
where \(g^{-}:=\min \{g(t,u),0\}\). Substituting (3.4) and (3.7) into (3.6), we have
$$\begin{aligned} \int^{\omega }_{0} \bigl\vert (Au)'(t) \bigr\vert ^{p}\,dt \leq & \bigl(1+\vert c\vert \bigr)\Vert u \Vert _{\infty } \bigl( 2a \omega \Vert u\Vert _{\infty }^{p-1}+2b \omega +2K\omega +\Vert e\Vert _{\infty } \omega \bigr) \\ = &2 \bigl(1+\vert c\vert \bigr)a\omega \Vert u\Vert _{\infty }^{p}+ \bigl(1+\vert c\vert \bigr)N_{1} \Vert u\Vert _{\infty } \\ \leq &2 \bigl(1+\vert c\vert \bigr)a\omega \biggl( D_{1}+ \frac{1}{2} \int^{\omega }_{0} \bigl\vert u'(t) \bigr\vert \,dt \biggr) ^{p} \\ &{}+ \bigl(1+\vert c\vert \bigr)N_{1} \biggl( D_{1}+\frac{1}{2} \int^{\omega }_{0} \bigl\vert u'(t) \bigr\vert \,dt \biggr) \\ = &\frac{(1+\vert c\vert )a\omega }{2^{p-1}} \biggl( 1+\frac{2D_{1}}{\int^{\omega }_{0}\vert u'(t)\vert \,dt} \biggr) ^{p} \biggl( \int^{\omega }_{0} \bigl\vert u'(t) \bigr\vert \,dt \biggr) ^{p} \\ &{}+ \frac{1}{2} \bigl(1+\vert c\vert \bigr)N_{1} \int^{\omega }_{0} \bigl\vert u'(t) \bigr\vert \,dt+ \bigl(1+\vert c\vert \bigr)N_{1}D _{1}, \end{aligned}$$
where \(N_{1}:=2b\omega +2K\omega +\Vert e\Vert _{\infty }\omega \). For a given constant \(\zeta >0\), which is only dependent on \(k>0\), we have
$$(1+u)^{k}\leq 1+(1+k)u,\quad \mbox{for } u\in [0,\zeta ]. $$
Therefore, we have
$$\begin{aligned} \int^{\omega }_{0} \bigl\vert (Au)'(t) \bigr\vert ^{p}\,dt \leq & \frac{(1+\vert c\vert )a\omega }{2^{p-1}} \biggl( 1+ \frac{2D_{1}p}{\int^{\omega } _{0}\vert u'(t)\vert \,dt} \biggr) \biggl( \int^{\omega }_{0} \bigl\vert u'(t) \bigr\vert \,dt \biggr) ^{p} \\ &{}+ \frac{1}{2} \bigl(1+\vert c\vert \bigr)N_{1} \int^{\omega }_{0} \bigl\vert u'(t) \bigr\vert \,dt+ \bigl(1+\vert c\vert \bigr)N_{1}D _{1} \\ = &\frac{(1+\vert c\vert )a\omega }{2^{p-1}} \biggl( \int^{\omega }_{0} \bigl\vert u'(t) \bigr\vert \,dt \biggr) ^{p}+\frac{(1+\vert c\vert )a\omega D_{1}p}{2^{p-2}} \biggl( \int^{\omega }_{0} \bigl\vert u'(t) \bigr\vert \,dt \biggr) ^{p-1} \\ &{}+\frac{1}{2} \bigl(1+\vert c\vert \bigr)N_{1} \int^{\omega }_{0} \bigl\vert u'(t) \bigr\vert \,dt+ \bigl(1+\vert c\vert \bigr)N_{1}D _{1}. \end{aligned}$$
(3.8)
By application of Lemma 2.1, we have
$$\begin{aligned} \int^{\omega }_{0} \bigl\vert u'(t) \bigr\vert \,dt = & \int^{\omega }_{0} \bigl\vert \bigl(A^{-1}Au' \bigr) (t) \bigr\vert \,dt \\ \leq &\frac{\int^{\omega }_{0}\vert (Au)'(t)\vert \,dt}{\vert 1-\vert c\vert \vert } \\ \leq & \frac{\omega^{\frac{1}{q}} ( \int^{\omega }_{0}\vert (Au)'(t)\vert ^{p}\,dt) ^{\frac{1}{p}}}{\vert 1-\vert c\vert \vert }, \end{aligned}$$
(3.9)
since \((Au')(t)=(Au)'(t)\) and \(\frac{1}{p}+\frac{1}{q}=1\). Apply the inequality
$$(a+b)^{k}\leq a^{k}+ b^{k},\quad \mbox{for } a, b>0, 0< k< 1. $$
Substituting (3.8) into (3.9), we have
$$\begin{aligned} \int^{\omega }_{0} \bigl\vert u'(t) \bigr\vert \,dt \leq & \frac{\omega^{\frac{1}{q}} ( \frac{(1+\vert c\vert )a \omega }{2^{p-1}} ) ^{\frac{1}{p}}\int^{\omega }_{0}\vert u'(t)\vert \,dt + \omega^{\frac{1}{q}} ( \frac{(1+\vert c\vert )a\omega D_{1}p}{2^{p-2}} ) ^{\frac{1}{p}} ( \int^{\omega }_{0}\vert u'(t)\vert \,dt) ^{\frac{p-1}{p}}}{\vert 1-\vert c\vert \vert } \\ &{}+\frac{\omega^{\frac{1}{q}} ( \frac{1}{2}(1+\vert c\vert )N_{1}\int^{ \omega }_{0}\vert u'(t)\vert \,dt) ^{\frac{1}{p}}+\omega^{\frac{1}{q}} ( (1+\vert c\vert )N _{1}D_{1} ) ^{\frac{1}{p}}}{\vert 1-\vert c\vert \vert }. \end{aligned}$$
Since \(\frac{\omega (1+\vert c\vert )^{\frac{1}{p}}a^{\frac{1}{p}}}{\vert 1-\vert c\vert \vert }<2^{ \frac{p-1}{p}}\), it is easy to see that there exists a positive constant \(M_{1}'\) such that
$$ \int^{\omega }_{0} \bigl\vert u'(t) \bigr\vert \,dt\leq M_{1}'. $$
(3.10)
From (3.4) and (3.10), we have
$$ \Vert u\Vert _{\infty }\leq D_{1}+ \frac{1}{2} \int^{\omega }_{0} \bigl\vert u'(t) \bigr\vert \,dt \leq D_{1}+\frac{1}{2}M_{1}':=M_{1}. $$
(3.11)
As \((Au)(0)=(Au)(\omega)\), there exists \(t_{1}\in [0,\omega ]\) such that \((Au)'(t_{1})=0\), while \(\phi_{p}(0)=0\), we have
$$\begin{aligned} \bigl\vert \phi_{p} \bigl((Au)'(t) \bigr) \bigr\vert = & \biggl\vert \int^{t}_{t_{1}} \bigl(\phi_{p} \bigl((Au)'(s) \bigr) \bigr)'\,ds \biggr\vert \\ \leq & \lambda \int^{\omega }_{0} \bigl\vert f \bigl(t,u'(t) \bigr) \bigr\vert \,dt+\lambda \int^{\omega }_{0} \bigl\vert g \bigl(t,u(t) \bigr) \bigr\vert \,dt+\lambda \int^{\omega }_{0} \bigl\vert e(t) \bigr\vert \,dt, \end{aligned}$$
(3.12)
where \(t\in [t_{1},t_{1} +\omega ]\). In view of \((H_{1})\), (3.7) and (3.12), we have
$$\begin{aligned} \bigl\Vert \phi_{p}(Au)' \bigr\Vert _{\infty } = &\max_{t\in [0,\omega ]} \bigl\{ \bigl\vert \phi_{p} \bigl((Au)'(t) \bigr) \bigr\vert _{\infty } \bigr\} \\ = &\max_{t\in [t_{1},t_{1}+\omega ]} \biggl\{ \biggl\vert \int^{t}_{t_{1}} \bigl(\phi _{p} \bigl((Au)'(s) \bigr) \bigr)'\,ds \biggr\vert \biggr\} \\ \leq &\lambda \biggl( \int^{\omega }_{0} \bigl\vert f \big(t,u'(t) \bigr\vert \,dt+ \int^{\omega } _{0} \bigl\vert g \bigl(t,u(t) \bigr) \bigr\vert \,dt+ \int^{\omega }_{0} \bigl\vert e(t) \bigr\vert \,dt \biggr) \\ \leq &\lambda \bigl( K\omega +2a\omega \Vert u\Vert _{\infty }^{p-1}+2b \omega +K\omega +\Vert e\Vert _{\infty }\omega \bigr) \\ \leq &\lambda \bigl( 2a\omega M_{1}^{p-1}+2K\omega +2b\omega +\Vert e\Vert _{\infty }\omega \bigr):=\lambda M_{2}'. \end{aligned}$$
(3.13)
We claim that there exists a positive constant \(M_{2}>M_{2}'+1\) such that, for all \(t\in \mathbb{R}\),
$$ \bigl\Vert u' \bigr\Vert _{\infty }\leq M_{2}. $$
(3.14)
In fact, if \(u'\) is not bounded, there exists a positive constant \(M_{2}''\) such that \(\Vert u'\Vert _{\infty }>M_{2}''\) for some \(u'\in \mathbb{R}\). Therefore, we have
$$\begin{aligned} \bigl\Vert \phi_{p}(Au)' \bigr\Vert _{\infty } =& \bigl\Vert \phi_{p} \bigl(Au' \bigr) \bigr\Vert _{\infty }= \bigl\Vert Au' \bigr\Vert _{ \infty }^{p-1} \\ =& \bigl(1+\vert c\vert \bigr)^{p-1} \bigl\Vert u' \bigr\Vert _{\infty }^{p-1}\geq \bigl(1+\vert c\vert \bigr)^{p-1}M _{2}^{\prime\prime \, p-1}:=M_{2}^{*}. \end{aligned}$$
Then it is a contradiction. So (3.14) holds.
On the other hand, it follows by (2.1) that
$$ \bigl(\phi_{p}(Au)'(t) \bigr)'+\lambda f \bigl(t,u'(t) \bigr)+\lambda \bigl(g_{0} \bigl(u(t) \bigr)+g_{1} \bigl(t,u(t) \bigr) \bigr)= \lambda e(t). $$
(3.15)
Multiplying both sides of (3.15) by \(u'(t)\) we get
$$\begin{aligned}& \bigl(\phi_{p}(Au)'(t) \bigr)'u'(t)+\lambda f \bigl(t,u'(t) \bigr)u'(t)+\lambda \bigl(g_{0} \bigl(u(t) \bigr)+g _{1} \bigl(t,u(t) \bigr) \bigr)u'(t) \\& \quad =\lambda e(t)u'(t). \end{aligned}$$
(3.16)
Let \(\tau \in [0,\omega ]\) be as in (3.3), for any \(\tau \leq t\leq \omega\), we integrate (3.16) on \([\tau,t]\) and get
$$\begin{aligned} \lambda \int^{u(t)}_{u(\tau)}g_{0}(v)\,dv = &\lambda \int^{t}_{\tau }g _{0} \bigl(u(s) \bigr)u'(s)\,ds \\ = &- \int^{t}_{\tau } \bigl(\phi_{p}(Au)'(s) \bigr)'u'(s)\,ds-\lambda \int^{t}_{ \tau }f \bigl(s,u'(s) \bigr)u'(s)\,ds \\ &{}-\lambda \int^{t}_{\tau }g_{1} \bigl(s,u(s) \bigr)u'(s)\,ds+\lambda \int^{t}_{ \tau }e(s)u'(s)\,ds. \end{aligned}$$
(3.17)
By (3.7), (3.11) and (3.14), we have
$$\begin{aligned} &\biggl\vert \int^{t}_{\tau } \bigl(\phi_{p}(Au)'(s) \bigr)'u'(s)\,ds \biggr\vert \\ &\quad \leq \int^{T} _{0} \bigl\vert \bigl( \phi_{p}(Au)'(s) \bigr)' \bigr\vert \bigl\vert u'(s) \bigr\vert \,ds \\ &\quad \leq \lambda \bigl\Vert u' \bigr\Vert _{\infty } \biggl( \int^{\omega }_{0}\big\vert f \bigl(t,u'(t) \bigr)\,dt+ \int^{\omega }_{0} \bigl\vert g \bigl(t,u(t) \bigr) \bigr\vert \,dt+ \int^{\omega }_{0} \bigl\vert e(t) \bigr\vert \,dt \biggr) \\ &\quad \leq \lambda M_{2} \bigl( K\omega +2a\omega \vert u\Vert _{\infty }^{p-1}+2b \omega +K\omega +\Vert e\Vert _{\infty }\omega \bigr) \\ &\quad \leq\lambda M_{2} \bigl( 2K\omega +2a\omega M_{1}^{p-1}+2b \omega + \Vert e\Vert _{\infty }\omega \bigr). \end{aligned}$$
Moreover, from \((H_{1})\) and (3.14)
$$\begin{aligned} & \biggl\vert \int^{t}_{\tau }f \bigl(s,u'(s) \bigr)u'(s)\,ds \biggr\vert \leq \int^{T}_{0} \bigl\vert f \bigl(s,u'(s) \bigr) \bigr\vert \bigl\vert u'(s) \bigr\vert \,ds \leq KM_{2} \omega, \\ & \biggl\vert \int^{t}_{\tau }g_{1} \bigl(s,u(s) \bigr)u'(s)\,ds \biggr\vert \leq \int^{T}_{0} \bigl\vert g_{1} \bigl(s,u(s) \bigr) \bigr\vert \bigl\vert u'(s) \bigr\vert \,ds \leq M_{2} \vert g_{M_{1}}\vert \sqrt{\omega }, \end{aligned}$$
where \(g_{M_{1}}=\max_{0\leq u\leq M_{1}}\vert g_{1}(t,u)\vert \in L^{2}(0, \omega)\).
$$\biggl\vert \int^{t}_{\tau }e(s)u'(s)\,ds \biggr\vert \leq \int^{\omega }_{0} \bigl\vert e(s) \bigr\vert \bigl\vert u'(s) \bigr\vert \,ds \leq \Vert e\Vert _{\infty } \omega M_{2}. $$
With these inequalities we can derive from (3.17) that
$$\biggl\vert \int^{u(t)}_{u(\tau)}g_{0}(v)\,dv \biggr\vert \leq M_{2} \bigl(3K\omega +2a \omega M_{1}^{p-1}+2b \omega +2\Vert e\Vert _{\infty }\omega +\vert g_{M_{1}}\vert \sqrt{ \omega } \bigr). $$
In view of (1.6), we know there exists \(M_{3}>0\) such that
$$ u(t)\geq M_{3},\quad \forall t\in [\tau,\omega ]. $$
(3.18)
The case \(t\in [0,\tau ]\) can be treated similarly.
Having in mind (3.11), (3.14) and (3.18), we define
$$\Omega = \bigl\{ u\in X:E_{1}< u(t)< E_{2} \mbox{ and } \bigl\vert u'(t) \bigr\vert < E_{3}\ \forall t\in \mathbb{R} \bigr\} , $$
where \(0< E_{1}< \min \{D_{2},M_{3}\}\), \(E_{2}>\max \{M_{1}, D_{1}\} \) and \(E_{3}>M_{2}\). We know that (2.1) has no solution on ∂Ω as \(\lambda \in (0,1)\) and when \(u(t)\in \partial \Omega \cap \mathbb{R}\), \(u(t)=E_{2}\) or \(u(t)=E_{1}\), from (3.4), we know that \(E_{2}+1>D_{1}\); therefore, from \((H_{2})\) we see that
$$\frac{1}{\omega } \int^{\omega }_{0}g(t,E_{2})\,dt< 0 $$
and
$$\frac{1}{\omega } \int^{\omega }_{0}g(t,E_{1})\,dt>0. $$
So condition (ii) is also satisfied. Set
$$H(u,\mu)=\mu u+(1-\mu)\frac{1}{\omega } \int^{\omega }_{0}g(t,u)\,dt, $$
where \(x\in \partial \Omega \cap \mathbb{R}\), \(\mu \in [0,1]\), we have
$$uH(u,\mu)=\mu u^{2}+(1-\mu)\frac{u}{\omega } \int^{\omega }_{0}g(t,u)\,dt \neq 0, $$
and thus \(H(u,\mu)\) is a homotopic transformation and
$$\begin{aligned} \deg \{F,\Omega \cap \mathbb{R},0\} &=\deg \biggl\{ \frac{1}{\omega } \int^{\omega }_{0}g(t,u)\,dt,\Omega \cap \mathbb{R},0 \biggr\} \\ &=\deg \{u,\Omega \cap \mathbb{R},0\}\neq 0. \end{aligned}$$
So condition (iii) is satisfied. In view of Lemma 2.1, there exists a solution with period ω. □