# Existence of periodic solutions for a class of *p*-Laplacian equations

- Xiaojun Chang
^{1, 2}Email author and - Yu Qiao
^{3}

**2013**:96

https://doi.org/10.1186/1687-2770-2013-96

© Chang and Qiao; licensee Springer 2013

**Received: **26 September 2012

**Accepted: **5 April 2013

**Published: **19 April 2013

## Abstract

This paper is devoted to the existence of periodic solutions for the one-dimensional *p*-Laplacian equation

where ${\varphi}_{p}(u)={|u|}^{p-2}u$ ($1<p<+\mathrm{\infty}$), $f\in C([0,2\pi ]\times \mathbb{R},\mathbb{R})$. By using some asymptotic interaction of the ratios $\frac{f(t,u)}{{|u|}^{p-2}u}$ and $\frac{p{\int}_{0}^{u}f(t,s)\phantom{\rule{0.2em}{0ex}}ds}{{|u|}^{p}}$ with the Fučík spectrum of $-{({\varphi}_{p}({u}^{\prime}))}^{\prime}$ related to periodic boundary condition, we establish a new existence theorem of periodic solutions for the one-dimensional *p*-Laplacian equation.

### Keywords

periodic solutions*p*-Laplacian Fučík spectrum Leray-Schauder degree Borsuk theorem

## 1 Introduction and main results

where ${\varphi}_{p}(u)={|u|}^{p-2}u$ ($1<p<+\mathrm{\infty}$), $f\in C([0,2\pi ]\times \mathbb{R},\mathbb{R})$. A solution *u* of problem (1.1) means that *u* is ${C}^{1}$ and ${\varphi}_{p}({u}^{\prime})$ is absolutely continuous such that (1.1) is satisfied for a.e. $t\in [0,2\pi ]$.

*p*-Laplacian have been obtained in the literature by many people (see [1–5]). Many solvability conditions for problem (1.1) were established by using the asymptotic interaction at infinity of the ratio $\frac{f(x,u)}{{|u|}^{p-2}u}$ with the Fučík spectrum for ${({\varphi}_{p}({u}^{\prime}))}^{\prime}$ under periodic boundary condition (see

*e.g.*, [2, 4, 6–9]). In [6], Del Pino, Manásevich and Murúa firstly defined the Fučík spectrum for ${({\varphi}_{p}({u}^{\prime}))}^{\prime}$ under periodic boundary value condition as the set ${\mathrm{\Sigma}}_{p}$ consisting of all the pairs $({\lambda}_{+},{\lambda}_{-})\in {\mathbb{R}}^{2}$ such that the equation

*π*-periodic solution (see [10] for $p=2$). Let

*f*is nonresonant with respect to the Fučík spectrum ${\mathrm{\Sigma}}_{p}$. In [11], Anane and Dakkak obtained a similar result by using the property of nodal set for eigenfunctions. If

*f*is resonant with respect to ${\mathrm{\Sigma}}_{p}$,

*i.e.*, there exists $({\lambda}_{+},{\lambda}_{-})\in {\mathrm{\Sigma}}_{p}$ such that ${lim}_{u\to +\mathrm{\infty}}\frac{f(t,u)}{{|u|}^{p-2}u}={\lambda}_{+}$, ${lim}_{u\to -\mathrm{\infty}}\frac{f(t,u)}{{|u|}^{p-2}u}={\lambda}_{-}$ uniformly for a.e. $t\in [0,2\pi ]$, together with the Landesman-Lazer type condition, Jiang [9] obtained the existence of solutions of (1.1) by applying the variational methods and symplectic transformations. In these works, either

*f*is resonant or nonresonant with respect to ${\mathrm{\Sigma}}_{p}$, the solvability of problem (1.1) was assured by assuming that the ratio $\frac{f(t,u)}{{|u|}^{p-2}u}$ stays at infinity in the pointwise sense asymptotically between two consecutive curves of ${\mathrm{\Sigma}}_{p}$. Note that

*p*-Laplacian equation

- (1)
There exist $b,{d}_{1},{d}_{2}>0$ such that ${d}_{1}\le \frac{g(u)}{{|u|}^{p-2}u}\le {d}_{2}$ for all $|u|\ge b$;

- (2)
${lim}_{u\to +\mathrm{\infty}}\frac{pG(u)}{{|u|}^{p}}={\lambda}_{+}$, ${lim}_{u\to -\mathrm{\infty}}\frac{pG(u)}{{|u|}^{p}}={\lambda}_{-}$ with $({\lambda}_{+},{\lambda}_{-})\notin {\mathrm{\Sigma}}_{p}$.

Here, the potential function *G* is nonresonant with respect to ${\mathrm{\Sigma}}_{p}$ and the ratio $\frac{g(u)}{{|u|}^{p-2}u}$ is not required to stay at infinity in the pointwise sense asymptotically between two consecutive branches of ${\mathrm{\Sigma}}_{p}$ and it may even cross at infinity multiple Fučík spectrum curves.

In this paper, we want to obtain the solvability of problem (1.1) by using the asymptotic interaction at infinity of both the ratios $\frac{f(x,u)}{{|u|}^{p-2}u}$ and $\frac{pF(t,u)}{{|u|}^{p}}$ with the Fučík spectrum for ${({\varphi}_{p}({u}^{\prime}))}^{\prime}$ under periodic boundary condition. Here, $F(t,u)={\int}_{0}^{u}f(t,s)\phantom{\rule{0.2em}{0ex}}ds$. The goal is to obtain the existence of solutions of (1.1) by requiring neither the ratio $\frac{f(x,u)}{{|u|}^{p-2}u}$ stays at infinity in the pointwise sense asymptotically between two consecutive branches of ${\mathrm{\Sigma}}_{p}$ nor the limits ${lim}_{u\to \pm \mathrm{\infty}}\frac{pF(t,u)}{{|u|}^{p}}$ exist. We shall prove that problem (1.1) admits a solution under the assumptions that the nonlinearity *f* has at most $(p-1)$-linear growth at infinity and the ratio $\frac{f(t,u)}{{|u|}^{p-2}u}$ has a ${L}^{1}$ limit as $u\to \pm \mathrm{\infty}$, while the ratio $\frac{pF(t,u)}{{|u|}^{p}}$ stays at infinity in the pointwise sense asymptotically between two consecutive branches of ${\mathrm{\Sigma}}_{p}$. Our result will complement the results in the literature on the solvability of problem (1.1) involving the Fučík spectrum.

For related works on resonant problems involving the Fučík spectrum, we also refer the interested readers to see [12–19] and the references therein.

Our main result for problem (1.1) now reads as follows.

**Theorem 1.1**

*Assume that*$f\in C([0,2\pi ]\times \mathbb{R})$

*and the following conditions hold*:

- (i)
*There exist constants*${C}_{1},M>0$*such that*$|f(t,u)|\le {C}_{1}(1+{|u|}^{p-1}),\phantom{\rule{1em}{0ex}}\mathit{\text{a.e.}}\phantom{\rule{1em}{0ex}}t\in [0,2\pi ],\mathrm{\forall}|u|\ge M;$(1.3) - (ii)
*There exists*${\eta}^{\pm}\in {L}^{\mathrm{\infty}}(0,2\pi )$*such that*${\int}_{0}^{2\pi}|\frac{f(t,u)}{{|u|}^{p-2}u}-{\eta}^{\pm}(t)|\phantom{\rule{0.2em}{0ex}}dt\to 0\phantom{\rule{1em}{0ex}}\mathit{\text{as}}\phantom{\rule{1em}{0ex}}u\to \pm \mathrm{\infty};$(1.4) - (iii)

*hold uniformly for a*.

*e*. $t\in [0,2\pi ]$

*with*

*Then problem* (1.1) *admits a solution*.

**Remark** If $f(t,u)=a(t){|u|}^{p-2}{u}^{+}-b(t){|u|}^{p-2}{u}^{-}+e(u)+h(t)$, where $a,b,h\in C[0,2\pi ]$ with ${p}_{1}\le a(t)\le {p}_{2}$, ${q}_{1}\le b(t)\le {q}_{2}$ and ${p}_{1},{p}_{2},{q}_{1},{q}_{2}>0$ satisfy (1.7), *e* is continuous on ℝ and ${lim}_{u\to +\mathrm{\infty}}\frac{e(u)}{{|u|}^{p-2}u}=0$, then ${lim}_{u\to +\mathrm{\infty}}\frac{pF(t,u)}{{|u|}^{p}}=a(t)$ and ${lim}_{u\to -\mathrm{\infty}}\frac{pF(t,u)}{{|u|}^{p}}=b(t)$. By Theorem 1.1, it follows that problem (1.1) admits a solution. It is easily seen that the result of [16] cannot be applied to this case. Note that one can also obtain the solvability of (1.1) in this case by the result of [6], while in Theorem 1.1 we do not require the pointwise limit at infinity of the ratio $\frac{f(t,u)}{{|u|}^{p-2}u}$ as in [6].

*m*-times continuous differential real functions with norm

## 2 Proof of the main result

Denote by deg the Leray-Schauder degree. To prove Theorem 1.1, we need the following results.

**Lemma 2.1** [20]

*Let* Ω *be a bounded open region in a real Banach space* *X*. *Assume that* $K:\overline{\mathrm{\Omega}}\to \mathbb{R}$ *is completely continuous and* $p\notin (I-K)(\partial \mathrm{\Omega})$. *Then the equation* $(I-K)(x)=p$ *has a solution in* Ω *if* $deg(I-K,\mathrm{\Omega},p)\ne 0$.

**Lemma 2.2** (Borsuk Theorem [20])

*Assume that* *X* *is a real Banach space*. *Let* Ω *be a symmetric bounded open region with* $\theta \in \mathrm{\Omega}$. *Assume that* $K:\overline{\mathrm{\Omega}}\to \mathbb{R}$ *is completely continuous and odd with* $\theta \notin (I-K)(\partial \mathrm{\Omega})$. *Then* $deg(I-K,\mathrm{\Omega},\theta )$ *is odd*.

*Proof of Theorem 1.1*Take $({\lambda}_{+},{\lambda}_{-})\in [{p}_{1},{p}_{2}]\times [{q}_{1},{q}_{2}]$. Consider the following homotopy problem:

where $\mu \in [0,1]$.

*u*is a solution of problem (2.1), then

*n*, we have ${\parallel {z}_{n}\parallel}_{{C}^{1}}\le {M}_{1}$, and thus there exists ${z}_{0}\in {C}^{1}[0,2\pi ]$ such that, passing to a subsequence if possible,

- (i)
${z}_{0}$ changes sign in $[0,2\pi ]$;

- (ii)
${z}_{0}(t)\ge 0$, $\mathrm{\forall}t\in [0,2\pi ]$;

- (iii)
${z}_{0}(t)\le 0$, $\mathrm{\forall}t\in [0,2\pi ]$.

In the following, it will be shown that each case leads to a contradiction.

Here, $\tilde{\alpha}(t)=(1-{\mu}_{0}){\lambda}_{+}+{\mu}_{0}{\eta}_{+}(t)$, $\tilde{\beta}(t)=(1-{\mu}_{0}){\lambda}_{-}+{\mu}_{0}{\eta}_{-}(t)$.

A contradiction. Hence, (2.9) holds.

holds uniformly for a.e. $t\in [0,2\pi ]$.

Then by (1.7), it follows that ${z}_{0}\equiv 0$. A contradiction. Combining (2.12)-(2.13) with (2.15)-(2.18), we obtain a contradiction.

By $\tilde{\alpha}(t)\ge {p}_{1}>0$ for a.e. $t\in [0,2\pi ]$, it follows that ${z}_{0}(t)=0$ for a.e. $t\in [0,2\pi ]$, which is contrary to that ${\parallel {z}_{0}\parallel}_{\mathrm{\infty}}=1$. Hence, (2.22) holds. Clearly, (2.21)-(2.22) contradict (2.19).

Case (iii). In this case, ${r}_{n}\to -\mathrm{\infty}$ and $\{{s}_{n}\}$ is uniformly bounded. Similar arguments as in Case (ii) imply a contradiction.

*u*is a solution of problem (2.1), then

Hence, problem (1.1) has a solution. The proof is complete. □

## Declarations

### Acknowledgements

The first author sincerely thanks Professor Yong Li and Doctor Yixian Gao for their many useful suggestions and the both authors thank Professor Zhi-Qiang Wang for many helpful discussions and his invitation to Chern Institute of Mathematics. The first author is partially supported by the NSFC Grant (11101178), NSFJP Grant (201215184) and FSIIP of Jilin University (201103203). The second author is partially sup- ported by NSFC Grant (11226123).

## Authors’ Affiliations

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