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Periodic solutions for nonlocal \(p(t)\)-Laplacian systems
Boundary Value Problems volume 2019, Article number: 119 (2019)
Abstract
The purpose of this paper is to investigate the existence of periodic solutions for a class of nonlocal \(p(t)\)-Laplacian systems. When the nonlinear term is \(p^{+}\)-superlinear at infinity, some new solvability conditions of nontrivial periodic solutions are obtained by using a version of the local linking theorem. A major point is that we ensure compactness without the well-known Ambrosetti–Rabinowitz type superlinearity condition. In addition, by applying the saddle point theorem, we established the existence of at least one periodic solution for such problems with a \(p^{-}\)-subquadratic potential.
1 Introduction and main results
Consider the non-autonomous second order Hamiltonian systems
where \(T>0\), \(u(t)\in \mathbb{R}^{N}\), \(\nabla V(t,u)\) denotes the gradient of \(V(t,u)\) in u.
In the classical monograph [1], Mawhin and Willem investigated the existence of periodic solutions for problem (1.1) via critical point theory. During the past two decade, inspire by [1] the existence and multiplicity of periodic solutions for systems (1.1) has been studied extensively.
Problems with variable exponent growth conditions arise in the description of the physical phenomena with “pointwise different properties” which first arose from the nonlinear elasticity theory; see [2]. It was also observed that non-homogeneous \(p(t)\)-Laplacian operators are related to modeling of so-called electrorheological fluids; see [3, 4] for more details of the physical aspects. Another field of application of \(p(t)\)-Laplacian systems is image processing [5, 6]. The variable nonlinearity is used to eliminate possible noise and outline the borders of the true image.
In [7,8,9,10,11,12,13,14,15], the authors studied the existence of periodic solutions, subharmonic solutions and homoclinic orbits for the \(p(t)\)-Laplacian systems
where \(p(t)\in C([0,T],\mathbb{R}^{+})\), the operator \(\frac{d}{dt}( \vert \dot{u}(t) \vert ^{p(t)-2}\dot{u}(t))\) is said to be \(p(t)\)-Laplacian. The \(p(t)\)-Laplacian operator possesses more complicated nonlinearity than that of the p-Laplacian operator, where \(p>1\) is a constant. For example, it is inhomogeneous, which provokes some mathematical difficulties. We point out that commonly known techniques for studying constant exponent equations fail in the setting of problems involving variable exponents.
In this paper, we consider the following nonlocal \(p(t)\)-Laplacian systems:
where \(T>0\), \(M(s):[0,+\infty )\rightarrow (0,+\infty )\) is a continuous function, and assume that \(V(t,u)\) satisfies the following assumption:
- \((V_{0})\) :
-
\(V(t,u)\) is measurable in t for each \(u\in \mathbb{R}^{N}\) and continuously differentiable in u for a.e. \(t\in [0,T]\), and there exist \(a\in C(\mathbb{R}^{+},\mathbb{R}^{+})\), \(b\in L^{1}([0,T];\mathbb{R}^{+})\) such that
$$\begin{aligned}& \bigl\vert V(t,u) \bigr\vert + \bigl\vert \nabla V(t,u) \bigr\vert \leq a\bigl( \vert u \vert \bigr) b(t), \end{aligned}$$for all \(u\in \mathbb{R}^{N}\) and a.e. \(t\in [0,T]\).
Throughout the paper, we assume that \(p(t)\) appearing in problem (1.3) satisfies
- \((P)\) :
-
\(p(t)\in C([0,T],\mathbb{R}^{+})\), \(p(t)=p(t+T)\) and \(p(t)\) fulfills the following hypothesis:
$$\begin{aligned}& 1< p^{-}: =\min_{0\leq t\leq T}p(t)\leq p^{+}: =\max _{0\leq t\leq T}p(t)< +\infty . \end{aligned}$$
Comparing with systems (1.1) or (1.2), one typical feature of the equation in problem (1.3) is the nonlocality. Since the presence of an integral term \(M (\int _{0}^{T} \frac{ \vert \dot{u}(t) \vert ^{p(t)}}{p(t)} \,dt )\), thus the equation in problem (1.3) is no longer a pointwise identity. From the physical point of view, the nonlocal coefficient \(M (\int _{0}^{T}\frac{ \vert \dot{u}(t) \vert ^{p(t)}}{p(t)} \,dt )\) is a function depending on the average of the kinetic energy. Alternatively, a parabolic version of problem (1.3) can model a spreading process of a particular species within the domain, where u is its population density; see [16]. During the past decade, much interest has grown on nonlocal problems with various boundary data. In the very recent paper [17], the authors studied an elliptic boundary value problem with degenerate nonlocal term, and they proved a multiplicity result of positive solutions for the problem, where the number of solutions doubles the number of zeros of the degenerate term. In [18], the authors investigated a class of Kirchhoff problem in a bounded domain, where the nonlocal coefficient to vanish in many different points, and the existence of multiple solutions is also established by using a priori estimates, variational methods and truncation techniques.
It is worth mentioning that there are some papers concerning related equations or abstract spaces with variable exponents. In the monograph [19], Radulescu and Repovs provided a thorough introduction to the theory of nonlinear partial differential equations (PDEs) with a variable exponent. In [20], the authors considered a class of double phase variational integrals driven by non-homogeneous potentials, and the analysis developed in this paper extends the abstract framework corresponding to some standard differential operator with variable exponent. In [21], Ho and Sim obtained the existence of solutions of weighted elliptic equations containing a convection term with variable exponents. Liu and Zhao in [22] introduced the definition of abstract Hardy spaces with variable exponents. In [23], the authors proved continuity of the flow and weak upper semicontinuity of a family of global attractors for reaction-diffusion equations with spatially variable exponents. In recent years, some scholars have investigated nonlocal \(p(x)\)-Laplacian equations with Dirichlet or Neumann boundary condition; see [24,25,26,27]. However, as far as we known, there are few papers discussing periodic solutions for nonlocal problems (1.3). For instance, by means of a variational analysis and the theory of variable exponent spaces, the author in [28] obtained the existence and multiplicity of periodic solutions for systems (1.3) with \(p^{-}\)-sublinear nonlinear term.
For \(p(t)\)-Laplacian systems, the well-known Ambrosetti–Rabinowitz type superlinearity condition (AR-condition, for short) is the following: there exist \(R>0\) and \(\mu >p^{+}\) such that
for \(\vert u \vert \geq R\) and \(t\in [0,T]\). This kinds of technical condition implies that \(\nabla V(t,u)\) grows at a \(p^{+}\)-superlinear rate as \(\vert u \vert \rightarrow +\infty \). The main role of AR-condition is to ensure the compactness required by minimax arguments, and without the AR-condition the situation is more complicated. However, there are many functions which are superlinear at infinity, but do not satisfy the AR-condition, and these functions have attracted much interest in recent years, for example; see [29,30,31,32,33].
In this paper, under no AR-condition, we study the existence of nontrivial periodic solutions for problem (1.3) with \(p^{+}\)-superlinear nonlinear terms at infinity.
In the sequel, we will assume that \(M(s):[0,+\infty )\rightarrow (0,+ \infty )\) is continuous and
- \((M_{0})\) :
-
there exists a constant \(m_{0}>0\) such that \(M(s) \geq m_{0}\), for all \(s\in [0,+\infty )\);
- \((M_{1})\) :
-
there exists a constant \(\eta \geq 1\) such that
$$\begin{aligned}& \widehat{M}(s):= \int ^{s}_{0}M(\sigma ) \,d\sigma \geq \frac{1}{\eta }M(s)s, \end{aligned}$$for all \(s\in [0,+\infty )\).
Now, we suppose that \(V:[0,T]\times \mathbb{R}^{N}\rightarrow \mathbb{R}\) satisfies the following assumption:
- (\(V_{1}\)):
-
\(V(t,u)\geq 0\), for all \(u\in \mathbb{R}^{N}\) and a.e. \(t\in [0,T]\);
- (\(V_{2}\)):
-
\(\limsup_{ \vert u \vert \rightarrow 0} \frac{ \vert V(t,u) \vert }{ \vert u \vert ^{p ^{+}}}\leq \frac{m_{0}}{2p^{+}TC_{0}^{p^{+}}}\), uniformly for a.e. \(t\in [0,T]\);
- (\(V_{3}\)):
-
there exists a decreasing function \(h\in C(\mathbb{R} ^{+},\mathbb{R}^{+})\), \(r>0\), such that
$$\begin{aligned}& \bigl(\eta p^{+}+h\bigl( \vert u \vert \bigr) \bigr) V(t,u)\leq \bigl(\nabla V(t,u),u \bigr), \end{aligned}$$
for a.e. \(t\in [0,T]\) and \(u\in \mathbb{R}^{N}\) with \(\vert u \vert \geq r\), where h satisfies the following properties:
- (\(h_{1}\)):
-
\(\lim_{ \vert u \vert \rightarrow +\infty } u h( \vert u \vert )=+\infty \), \(\forall u\in \mathbb{R}^{+}\);
- (\(h_{2}\)):
-
\(\lim_{ \vert u \vert \rightarrow +\infty } H( \vert u \vert )=+\infty \), where \(H( \vert u \vert )= \exp (\int _{r}^{ \vert u \vert }\frac{h(\sigma )}{\sigma }\,d\sigma )\).
- (\(V_{4}\)):
-
\(\liminf_{ \vert u \vert \rightarrow +\infty }\frac{V(t,u)}{ \vert u \vert ^{ \eta p^{+}} H( \vert u \vert )}\geq \kappa >0\), uniformly for a.e. \(t\in [0,T]\).
Our main result is the following theorem.
Theorem 1.1
Assume that conditions (P), (\(M _{0}\)), (\(M_{1}\)) and (\(V_{0}\))–(\(V_{4}\)) are satisfied. Then problem (1.3) has at least one nontrivial T-periodic solution.
Remark 1.1
Take the nonlocal coefficient
then M satisfies (\(M_{0}\)) and (\(M_{1}\)) with \(m_{0}=1\) and \(\eta =1\), let
where \(f(t)\in L^{1}([0,T];\mathbb{R}^{+})\) with \(\inf_{t\in [0,T]}f(t)>0\). Hence, we obtain
Then all the conditions of Theorem 1.1 hold, and V is not covered by results in [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26].
Next, we obtain the existence of periodic solutions for problem (1.3) with \(p^{-}\)-subquadratic potential. Now, we state the assumptions on function M and V:
- \((M_{2})\) :
-
$$\begin{aligned}& \widehat{M}(s):= \int ^{s}_{0}M(\sigma ) \,d\sigma \leq M(s)s, \end{aligned}$$
for all \(s\in [ 0,+\infty )\).
- \((V_{5})\) :
-
There exists a function θ with \(0<\frac{1}{ \theta ( \vert u \vert )}<p^{-}\), \(L>0\) such that
$$\begin{aligned}& \bigl(\nabla V(t,u),u \bigr)\leq \biggl(p^{-}-\frac{1}{\theta ( \vert u \vert )} \biggr)V(t,u), \end{aligned}$$for all \(u\in \mathbb{R}^{N}\) with \(\vert u \vert \geq L\) and a.e. \(t\in [0,T]\), where θ satisfies the following properties:
- \((\theta _{1})\) :
-
\(\theta \in C(\mathbb{R}^{+},\mathbb{R}^{+})\);
- \((\theta _{2})\) :
-
\(\int _{L}^{\xi } \frac{1}{s\theta (s)} \,ds \rightarrow +\infty \) as \(\xi \rightarrow +\infty \).
- \((V_{6})\) :
-
\(V(t,u)\geq 0\) as \(\vert u \vert \rightarrow +\infty \) uniformly for a.e. \(t\in [0,T]\).
- \((V_{7})\) :
-
\(\int _{0}^{T}\frac{V(t,u)}{\theta ( \vert u \vert )} \,dt\rightarrow +\infty \) as \(\vert u \vert \rightarrow +\infty \).
Theorem 1.2
Assume that conditions (P), \((M_{0})\), \((M_{2})\), \((V_{0})\), \((V_{5})\), \((V_{6})\) and \((V_{7})\) are satisfied. Then problem (1.3) has at least one T-periodic solution.
Remark 1.2
When \(M(s)\equiv 1\), \(p(t)\equiv 2\), condition \((V_{5})\) was introduced in Wang and Xiao [27], which is an extension of the usual subquadratic growth condition. Theorem 1.2 generalizes Theorem 1.1 of Wang and Xiao [34] to the nonlocal and variable exponent space setting.
Remark 1.3
When \(M(s)\equiv 1\), Wang and Yuan [9] prove that problem (1.1) admits a T-periodic solution when the function \(V(t,u)\) is \(p^{-}\)-subquadratic in Rabinowitz’s sense, that is, there exists \(L>0\), \(0<\mu <p^{-}\) such that
for all \(\vert u \vert \geq L\) and a.e. \(t\in [0,T]\).
Define \(\inf_{ \vert u \vert \geq L} \frac{1}{\theta ( \vert u \vert )}: =\mathcal{K}\), where \(\mathcal{K}\) is a constant. If \(\mathcal{K}>0\), then condition \((V_{5})\) and (1.5) are equivalent. However, condition \((V_{5})\) is weaker than (1.5) when \(\mathcal{K}=0\).
Remark 1.4
The following example is presented for Theorem 1.2. For \(s\in [0,+\infty )\), let
where A, B and m are positive constants. Then the nonlocal coefficient M satisfies (\(M_{0}\)) and (\(M_{2}\)) with \(m_{0}=A\). Assume that \(p(t)\equiv \sin \frac{2\pi t}{T}+5\), then \(p^{-}=4\), let
where
Setting \(\theta ( \vert u \vert )=\ln (1+ \vert u \vert ^{4})\), then \(V(t,u)\) satisfies \((V_{5})\) but not (1.5).
2 Preliminaries
In this section, we recall some properties of the variable exponent Sobolev space.
Definition 2.1
([8])
Let \(p(t)\in C([0,T],\mathbb{R}^{+})\), and \(p(t)\) satisfies \((P)\). Define
with the norm
Definition 2.2
([8])
Define
Let \(u\in L^{1}([0,T],\mathbb{R}^{N})\) and \(v\in L^{1}([0,T], \mathbb{R}^{N})\). If
then v is called the T-weak derivative of u and is denoted by u̇.
Definition 2.3
([8])
Define
When \(p^{-}>1\), \(W_{T}^{1,p(t)}\) is reflexive Banach space, with the norm
For \(u\in W_{T}^{1,p(t)}\), we write
then \(\Vert \cdot \Vert \) is an equivalent norm on \(W_{T}^{1,p(t)}\), where \(\overline{u}=\frac{1}{T}\int _{0}^{T}u(t) \,dt\).
Lemma 2.1
([8])
If we denote
then
-
(i)
\(\vert u \vert _{p(t)}<1\ (=1;>1)\Leftrightarrow \rho (u)<1\ (=1;>1)\);
-
(ii)
\(\vert u \vert _{p(t)}>1\Rightarrow \vert u \vert _{p(t)}^{p^{-}}\leq \rho (u) \leq \vert u \vert _{p(t)}^{p^{+}}\);
-
(iii)
\(\vert u \vert _{p(t)}<1\Rightarrow \vert u \vert _{p(t)}^{p^{+}}\leq \rho (u) \leq \vert u \vert _{p(t)}^{p^{-}}\).
Lemma 2.2
([7])
There is a continuous embedding \(W_{T}^{1,p(t)}\hookrightarrow C([0,T],\mathbb{R}^{N})\), when \(p^{-}>1\), the embedding is compact. Then there exists \(C_{0}>0\), such that
Lemma 2.3
([9])
Let
then
-
(i)
\(\Vert u \Vert <(=;>)\,1\Leftrightarrow \varPhi (u)<(=;>)\,1\);
-
(ii)
\(\Vert u \Vert >1\Rightarrow \Vert u \Vert ^{p^{-}}\leq \varPhi (u)\leq \Vert u \Vert ^{p^{+}}\);
-
(iii)
\(\Vert u \Vert <1\Rightarrow \Vert u \Vert ^{p^{+}}\leq \varPhi (u)\leq \Vert u \Vert ^{p^{-}}\).
Lemma 2.4
([9])
\(J'\) is a bounded linear functional and a mapping of (\(S_{+}\)) on \(W_{T}^{1,p(t)}\), that is, if \(u_{n}\rightharpoonup u\) weakly in \(W_{T}^{1,p(t)}\) and \(\limsup_{n\rightarrow \infty } (J'(u_{n})-J'(u),u_{n}-u) \leq 0\), then \(\{u_{n}\}\) has a convergent subsequence, where \(J'\) is given by \(\langle J'(u),v\rangle =\int _{0}^{T}( \vert \dot{u}(t) \vert ^{p(t)-2}\dot{u}(t), \dot{v}(t)) \,dt\).
For \(u\in W_{T}^{1,p(t)}\), we can write \(u(t)=\bar{u}+\tilde{u}(t)\), where \(\bar{u}=\frac{1}{T}\int _{0}^{T}u(t) \,dt\). Let
It is easy to know \(\widetilde{W}_{T}^{1,p(t)}\) is a subset of \(W_{T}^{1,p(t)}\) and \(W_{T}^{1,p(t)}=\widetilde{W}_{T}^{1,p(t)}\oplus \mathbb{R}^{N}\).
Applying Lemma 2.1, from (2.1), it is easy to prove that
Lemma 2.5
For all \(\widetilde{u}\in \widetilde{W} _{T}^{1,p(t)}\), we have
-
(i)
\(\Vert \widetilde{u} \Vert >1\Rightarrow \Vert \widetilde{u} \Vert ^{p^{-}}\leq \int _{0}^{T} \vert \dot{u}(t) \vert ^{p(t)} \,dt\leq \Vert \widetilde{u} \Vert ^{p^{+}}\);
-
(ii)
\(\Vert \widetilde{u} \Vert <1\Rightarrow \Vert \widetilde{u} \Vert ^{p^{+}}\leq \int _{0}^{T} \vert \dot{u}(t) \vert ^{p(t)} \,dt \leq \Vert \widetilde{u} \Vert ^{p^{-}}\);
-
(iii)
\(\Vert \widetilde{u} \Vert =1\Rightarrow \int _{0}^{T} \vert \dot{u}(t) \vert ^{p(t)} \,dt=1\).
Lemma 2.6
For all \(\tilde{u}\in \widetilde{W}_{T} ^{1,p(t)}\), we have
3 Proof of Theorem 1.1
As usual, a weak solution of problem (1.3) is a function \(u\in W_{T} ^{1,p(t)}\) such that
holds for any \(v\in W_{T}^{1,p(t)}\).
The Euler–Lagrange functional associated to problem(1.3) given by
where
Then \(I\in C^{1}(W_{T}^{1,p(t)},\mathbb{R})\) whose Gateaux derivative is
for all \(u,v\in W_{T}^{1,p(t)}\). So we know that T-periodic weak solutions of problem (1.3) correspond to the critical points of the functional I.
A basic tool in this paper is the following abstract local linking theorem.
Theorem 3.1
([35])
Let X be a Banach space such that \(X=Y\oplus W\) with \(\dim Y<+\infty \). Assume that \(I\in C^{1}(X,\mathbb{R})\) satisfies the following conditions:
- \((I_{1})\) :
-
I satisfies the \((C)\) condition, that is, sequence \(\{u_{n}\}\subset X\) such that \(\{I(u_{n})\}\) is bounded and \(\Vert I'(u_{n}) \Vert (1+ \Vert u_{n} \Vert ) \rightarrow 0\) as \(n\rightarrow \infty \) has a convergent sequence.
- \((I_{2})\) :
-
I has a local linking at zero, that is, there exists a positive constant δ such that
$$\begin{aligned}& I(u)\leq 0, \quad \forall u\in Y, \Vert u \Vert \leq \delta ; \end{aligned}$$and
$$\begin{aligned}& I(u)\geq 0, \quad \forall u\in W, \Vert u \Vert \leq \delta . \end{aligned}$$ - \((I_{3})\) :
-
I maps bounded sets into bounded sets.
- \((I_{4})\) :
-
for every finite-dimensional subspace \(\widetilde{E}\subseteq W\), we have
$$\begin{aligned}& I(u)\rightarrow -\infty \quad \textit{as } \Vert u \Vert \rightarrow +\infty \textit{ and } u\in Y\oplus \widetilde{E}. \end{aligned}$$
Then I admits at least one nontrivial critical point.
Remark 3.1
The critical point theorem was originally due to Li and Willem [36]. Subsequently, Gasiński and Papageorgiou reformulate this result in the special case given in [35, Theorem 2.1].
In the sequel, we will denote various positive constants as \(D_{i}\) (\(i=1,2,3,\ldots \)).
Lemma 3.1
Suppose \((P)\), \((M_{0})\), \((M_{1})\), \((V_{0})\), \((V_{1})\), \((V_{3})\) and \((V_{4})\) hold. Then I satisfies the \((C)\) condition, that is, sequence \(\{u_{n}\}\subset W_{T}^{1,p(t)}\) such that \(\{I(u_{n})\}\) is bounded and \(\Vert I'(u_{n}) \Vert (1+ \Vert u_{n} \Vert ) \rightarrow 0\) as \(n\rightarrow +\infty \) has a convergent sequence.
Proof
Let \(\{u_{n}\}\subset W_{T}^{1,p(t)}\), such that \(\{I(u_{n})\}\) is bounded and \((1+ \Vert u_{n} \Vert ) \Vert I'(u_{n}) \Vert \rightarrow 0\) as \(n\rightarrow \infty \). There then exists a constant \(D_{1}>0\) such that
for all n. We claim that the sequence \(\{u_{n}\}\) is bounded in \(W^{1,p(t)}_{T}\). Suppose that is not the case, passing to a subsequence if necessary, we may assume that
Note that \(p^{+}:=\max_{0\leq t\leq T}p(t)\), using hypothesis (\(M _{1}\)), one has
Set
and
It follows from (\(V_{0}\)), (\(V_{1}\)) and (\(V_{3}\)) that
By (3.1), (3.3), (3.4) and (\(M_{0}\)), we have
which implies that
Combining (2.3) with (3.5), we have
For \(\forall u\in W_{T}^{1,p(t)}\), let \(\overline{u}=\frac{1}{T}\int _{0}^{T}u(t)\,dt\in \mathbb{R}^{N}\), \(\widetilde{u}(t)=u(t)- \overline{u}\). Set
then \(\{\omega _{n}\}\) is bounded in \(W^{1,p(t)}_{T}\) and \(\Vert \omega _{n} \Vert =1\). Hence, up to a subsequence, we get
and
Dividing both side of (3.6) by \(\Vert u_{n} \Vert ^{p^{-}}\), from (3.2) and (3.6), we find that
So we deduce that \(\omega =\overline{\omega }\in \mathbb{R}^{N}\) and \(\omega \neq 0\). Then, by (3.2), we infer that
uniformly for a.e. \(t\in [0,T]\). By (\(V_{1}\)), (\(V_{4}\)), (\(h_{2}\)), (3.8) and Fatou’s lemma, we get
Set \(s_{1}>0\), by condition (\(M_{1}\)), we obtain
for every \(s\in [s_{1},+\infty )\), where \(\eta \geq 1\). Integrating the above inequality we obtain
for every \(s\in [s_{1},+\infty )\), where \(\widehat{M}(s)=\int ^{s}_{0}M( \sigma ) \,d\sigma \). Therefore,
for every \(s\in [s_{1},+\infty )\). Thus there exist constants \(m_{1}>0\) and \(m_{2}>0\), such that
for all \(s>0\), where \(m_{1}: =\frac{\widehat{M}(s_{1})}{s_{1}^{ \eta }}\) and \(m_{2}: =\max_{s\in [0,s_{1}]}\widehat{M}(s)\).
According to (3.5) and (3.10), it suffices to show that
It follows from (3.1), (3.2), (3.11) and (\(V_{1}\)) that
which contradicts (3.9). Hence, \(\{u_{n}\}\) is bounded in \(W^{1,p(t)} _{T}\).
Notice that the space \(W_{T}^{1,p(t)}\) is reflexive, the sequence \(\{u_{n}\}\subset W_{T}^{1,p(t)}\) has a subsequence, also denoted by \(\{u_{n}\}\), such that
and
and \(\Vert u_{n} \Vert _{\infty }\leq D_{5}\) by (2.2), where \(D_{5}\) is a positive constant. So we have
where \(a_{0}=\max_{0\leq s\leq D_{5}}a(s)\). By the assumption \((1+ \Vert u_{n} \Vert ) \Vert I'(u_{n}) \Vert \rightarrow 0\) as \(n\rightarrow +\infty \), we have
Thus
Let \(J':W_{T}^{1,p(t)}\rightarrow (W_{T}^{1,p(t)})^{*}\) defined by
Then, by (\(M_{0}\)), we get \(\lim_{n\rightarrow +\infty }(J'(u _{n}),u_{n}-u)=0\). Furthermore, since \(J'(u)\) is bounded linear function, we get \(\lim_{n\rightarrow +\infty } (J'(u),u_{n}-u)=0\). Thus
Hence, using the (\(S_{+}\)) property (see Lemma 2.4), \(J'\) is a bounded linear functional and a mapping of (\(S_{+}\)) on \(W_{T}^{1,p(t)}\), it follows that \(\{u_{n}\}\) has a convergent subsequence. So we see that \(\{u_{n}\}\) admits a convergent sequence. Thus I satisfies condition \((C)\). □
Now we prove our main result Theorem 1.1.
Proof of Theorem 1.1
By Lemma 3.1 we have I satisfies (\(I_{1}\)) of Theorem 3.1, so it suffices to prove (\(I_{2}\)), (\(I_{3}\)) and (\(I_{4}\)) of Theorem 3.1.
-
Claim 1 Let \(W_{T}^{1,p(t)}=\widetilde{W}_{T}^{1,p(t)} \oplus \mathbb{R}^{N}\), we claim that I has a local linking at zero.
On the one hand, in view of condition \((V_{2})\), there exist two constants ρ and ε such that
where \(C_{0}\) is the same as in (2.2), and
for a.e. \(t\in [0,T]\) and \(\vert u \vert \leq \rho \). Let \(\delta :=\rho /C_{0}\), then \(\delta <1\). If \(\Vert u \Vert \leq \delta \), we have
By (\(M_{0}\)), (3.13), (3.14) and Lemma 2.5, for \(u\in \widetilde{W} _{T}^{1,p(t)}\) with \(\Vert u \Vert \leq \delta \), one has
This implies that \(I(u)\geq 0\), for \(u\in \widetilde{W}_{T}^{1,p(t)}\) with \(\Vert u \Vert \leq \delta \).
On the other hand, by (\(V_{1}\)), for all \(y\in \mathbb{R}^{N}\), one has
Hence, I has a local linking at zero.
-
Claim 2 We claim that I maps bounded sets into bounded sets.
For some positive constant \(D_{6}\), assume that \(\Vert u \Vert \leq D_{6}\), by (3.10), Lemma 2.3 and (\(V_{0}\)), we have
Hence, we deduce that I maps bounded sets into bounded sets.
-
Claim 3 Finally, we claim that
$$\begin{aligned}& I(u)\rightarrow -\infty \quad \text{as } \Vert u \Vert \rightarrow +\infty , \text{for } u\in \mathbb{R}^{N}\oplus \widetilde{E}, \end{aligned}$$where \(\widetilde{E}\subseteq \widetilde{W}_{T}^{1,p(t)}\) is a finite-dimensional linear subspace.
From (\(V_{4}\)), there exists a constant \(R_{1}>0\), such that
for \(u\in \mathbb{R}^{N}\) with \(\vert u \vert \geq R_{1}\) and a.e. \(t\in [0,T]\). Moreover, by (\(V_{0}\)) and (3.15), we can obtain
for all \(u\in \mathbb{R}^{N}\) and a.e. \(t\in [0,T]\), where \(D_{7}= \max_{ \vert u \vert \leq R_{1}}a( \vert u \vert )\int _{0}^{T}b(t)\,dt\).
Note that all norms on Ẽ are equivalent, there exist constants \(D_{8}>0\), such that
for all \(u\in \widetilde{E}\) and a.e. \(t\in [0,T]\). Combining (3.16) with (3.17), we have
for all \(u\in \widetilde{E}\) and a.e. \(t\in [0,T]\).
By (3.8), (3.18) and use Lemma 2.3, choosing \(\Vert u \Vert >1\), we have
From (3.17) and (\(h_{2}\)), one has
Therefore, we can infer that
Now, Theorem 1.1 is proved by Claims 1–3, Lemma 3.1 and Theorem 3.1. So problem (1.1) has at least one nontrivial T-periodic solution. □
4 Proof of Theorem 1.2
In this section, we will denote various positive constants as \(C_{i}\) (\(i=0,1,2,\ldots \)).
Lemma 4.1
Assume that assumptions \((P)\), \((M_{0})\), \((M_{2})\), (\(V_{0}\)), \((V_{5})\) and \((V_{7})\) hold, then the functional I satisfies the compactness condition \((C)\).
Proof
Suppose that \(\{u_{n}\}\) be a Cerami sequence in \(W_{T}^{1,p(t)}\), such that \(\{I(u_{n})\}\) is bounded and \((1+ \Vert u _{n} \Vert ) \Vert I'(u_{n}) \Vert \rightarrow 0\) as \(n\rightarrow \infty \). There then exists a constant \(C_{1}>0\) such that
for all \(n\in \mathbb{N}\). By assumption (\(V_{0}\)) and \((V_{5})\), one has
for a.e. \(t\in [0,T]\) and all \(u\in \mathbb{R}^{N}\), where \(d(t)=(p ^{-}+L)\max_{ \vert u \vert \leq L}a( \vert u \vert )b(t)\).
By condition (\(M_{2}\)), we have
It follows from (4.2) and (4.3) that
Hence we have
Define
By (\(V_{6}\)), we have
Then deduce from (\(V_{5}\))
for all \(s\geq \frac{L}{ \vert x \vert }\). Let
By solving the above equation, we obtain
where
which implies that
Therefore,
By assumption (\(V_{0}\)), we have
for a.e. \(t\in [0,T]\) and all \(x\in \mathbb{R}^{N}\). Thus, we get
for a.e. \(t\in [0,T]\) and all \(x\in \mathbb{R}^{N}\). Furthermore, by \((\theta _{2})\), we obtain
From \(0<\frac{1}{\theta (\tau )}<p^{-}\), for \(\tau >0\), we have
which implies that \(\tau ^{p^{-}}G(\tau )\) is increasing on τ.
Combining (4.6) and (2.2), we get
For \(\forall u\in W_{T}^{1,p(t)}\), let
By (\(M_{0}\)) and (2.3), we obtain
By (4.1), (4.7) and (4.8), we have
We claim that the sequence \(\{u_{n}\}\) is bounded in \(W^{1,p(t)}_{T}\). Suppose that is not the case. Passing to a subsequence if necessary, we may assume that
Set
then \(\{\omega _{n}\}\) is bounded in \(W^{1,p(t)}_{T}\) and \(\Vert \omega _{n} \Vert =1\). Hence, up to a subsequence, we get
and
By using Eqs. (4.6) and (4.10), one has
Dividing both sides of (4.9) by \(\Vert u_{n} \Vert ^{p^{-}}\), by (4.10) and (4.11), we find that
So we deduce that \(\omega =\overline{\omega }\in \mathbb{R}^{N}\) and \(\omega \neq 0\). Thus
for a.e. \(t\in [0,T]\). By \((V_{7})\), we get
This contradicts (4.4). Therefore \(\{u_{n}\}\) is bounded in \(W^{1,p(t)}_{T}\). Similar to the proof of Lemma 3.1, we see that I satisfies condition \((C)\). □
Proof of Theorem 1.2
By saddle point theorem (see Theorem 4.6 in [37]), we only need to verify the following linking conditions:
- (\(I_{1}\)):
-
\(I(u)\rightarrow +\infty \) as \(\Vert u \Vert \rightarrow + \infty \) in \(\widetilde{W}_{T}^{1,p(t)}\), where \(\widetilde{W}_{T} ^{1,p(t)}= \{u\in W_{T}^{1,p(t)}\mid \bar{u}=0 \}\).
- (\(I_{2}\)):
-
\(I(u)\rightarrow -\infty \) as \(\vert u \vert \rightarrow +\infty \) in \(\mathbb{R}^{N}\).
On one hand, by (\(M_{0}\)), (2.3) and (4.7), for all \(u\in \widetilde{W}_{T}^{1,p(t)}\), we have
Hence, by (4.11), we have
for all \(u\in \widetilde{W}_{T}^{1,p(t)}\).
On the other hand, since \(0<\frac{1}{\theta ( \vert x \vert )}<p^{-}\), by (\(V_{6}\)) and (\(V_{7}\)), we have
Now, the proof of Theorem 1.2 is completed. □
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Zhang, S. Periodic solutions for nonlocal \(p(t)\)-Laplacian systems. Bound Value Probl 2019, 119 (2019). https://doi.org/10.1186/s13661-019-1236-7
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DOI: https://doi.org/10.1186/s13661-019-1236-7