Abstract
In this paper, a n-dimensional prescribed mean curvature Rayleigh p-Laplacian equation with a deviating argument, \((\varphi_{p}(\frac{u'(t)}{\sqrt{1+|u'(t)|^{2}}}))'+F(t,u'(t))+G(t,u(t-\tau(t)))=e(t)\), is studied. By means of Mawhin’s continuation theorem and some analysis methods, a new result on the existence of homoclinic solutions for the equation is obtained. Our research enriches the contents of prescribed mean curvature equations.
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1 Introduction
In recent years, the existence of homoclinic solutions has been studied widely for the Hamiltonian systems and the p-Laplacian systems (see [1–4] and the references therein). For example, in [1], Lzydorek and Janczewska studied the existence of homoclinic solutions for second-order Hamiltonian system in the following form:
where \(q\in\mathbb{R}^{n} \) and \(V\in C^{1}(\mathbb{R}\times\mathbb {R}^{n},\mathbb{R})\), \(V(t,q)=-K(t,q)+W(t,q) \) is T-periodic with respect to t. Lu in [4] studied the existence of homoclinic solutions for a second-order p-Laplacian differential system with delay
where \(p\in(1,+\infty)\), \(\varphi_{p}:\mathbb{R}^{n}\rightarrow\mathbb {R}^{n}\), \(\varphi_{p}(u)=(|u_{1}|^{p-2}u_{1}, |u_{2}|^{p-2}u_{2},\ldots,|u_{n}|^{p-2}u_{n})\) for \(u\neq0=(0,0,\ldots,0)\) and \(\varphi_{p}(0)=(0,0,\ldots,0)\), \(F\in C^{2}(\mathbb{R}^{n},\mathbb{R})\), \(G, H\in C^{1}(\mathbb{R}^{n},\mathbb{R})\), \(e\in C(\mathbb{R}, \mathbb{R}^{n})\), and \(\gamma(t)\) is a continuous T-periodic function with \(\gamma(t)\geq0\); T is a given constant.
In the recent past, the prescribed mean curvature equation
and its modified forms, which arises from some problems associated to differential geometry and combustible gas dynamics, were studied extensively [5–10]. Also, we note that the existence of periodic solutions for the prescribed curvature mean equations has attracted much attention from researchers. For example, Feng in [11] studied the problem of the existence of periodic solution for a prescribed mean curvature Liénard equation
where \(\tau,e \in C(\mathbb{R},\mathbb{R}) \) are T-periodic, and \(g\in C(\mathbb{R}\times\mathbb{R},\mathbb{R})\) is T-periodic in the first argument, \(T>0\) is a constant. Aiming to apply Mawhin’s continuation theorem, Feng made (1.1) equivalent to the following system through the transformation \(v(t)=\frac{u'(t)}{\sqrt{1+(u'(t))^{2}}}\):
Li in [12] further studied the existence of periodic solutions for a prescribed mean curvature Rayleigh equation of the form
and Wang in [13] discussed the following boundary valued problem:
where \(p>1\) and \(\varphi_{p}: \mathbb{R}\rightarrow\mathbb{R}\) is given by \(\varphi_{p}(s)=|s|^{p-2}s\) for \(s\neq0\) and \(\varphi _{p}(0)=0\), \(g\in C(\mathbb{R}^{2},\mathbb{R})\), \(e, \tau\in C(\mathbb {R},\mathbb{R})\), \(g(t+\omega,x)=g(t,x)\), \(f(t+\omega,x)=f(t,x)\), \(f(t,0)=0\), \(e(t+\omega)=e(t)\) and \(\tau(t+\omega)=\tau(t)\). By using a similar transformation in [11], (1.2) is converted to the following system:
Under the conditions imposed on f and g such as
and
where \(a, r\geq1\); \(m_{1}\) and \(m_{2}\) are positive constants, the author found that (1.2) has at least one periodic solution. It is easy to see from the first equation of (1.3) that the function \(\varphi _{q}(x_{2}(t))\) must satisfy \(\max_{t\in[0,T]}|\varphi _{q}(x_{2}(t))|<1\). This implies that the open and bounded set Ω associated to Mawhin’s continuation theorem must satisfy \(\overline {\Omega} \in\{(x_{1}; x_{2})^{\top}\in X : \| x_{1}\| _{\infty} < M;\| x_{2}\|_{\infty} < 1\}\). Thus, there must be a constant \(\rho\in(0,1)\) such that \(\Omega\in\{(x_{1}; x_{2})^{\top}\in X : \| x_{1}\|_{\infty} < M;\| x_{2}\|_{\infty} < \rho\}\). But in [13], the author obtained \(\Omega= \{(x_{1}; x_{2})^{\top}\in X : \| x_{1}\|_{\infty} < M_{1};\| x_{2}\|_{\infty } <M_{2}\} \) and there was no proof as regards \(M_{2}<1\). A similar problem also occurred in [14].
Inspired by the above fact, the aim of this paper is to investigate the existence of homoclinic solution to the following n-dimensional prescribed mean curvature equation with a deviating argument:
where \(p\in(1,+\infty)\), \(\varphi_{p}:\mathbb{R}^{n}\rightarrow\mathbb {R}^{n}\), \(\varphi_{p}(u)=(|u_{1}|^{p-2}u_{1}, |u_{2}|^{p-2}u_{2},\ldots,|u_{n}|^{p-2}u_{n})\) for \(u\neq0=(0,0,\ldots,0)\) and \(\varphi_{p}(0)=(0,0,\ldots,0)\), \(F\in C(\mathbb{R}\times\mathbb {R}^{n};\mathbb{R}^{n})\), \(G\in C (\mathbb{R}\times\mathbb {R}^{n};\mathbb{R}^{n} )\), \(e\in C(\mathbb{R};\mathbb{R}^{n})\), \(\tau(t)\) is a continuous T-periodic function and \(T > 0\) is given constant.
In order to study the homoclinic solution for (1.4), firstly, like in [1–4, 15] and [16], the existence of a homoclinic solution for (1.4) is obtained as a limit of a certain sequence of \(2kT\)-periodic solutions for the following equation:
where \(k\in\mathbb{N}\). \(e_{k} : \mathbb{R}\rightarrow\mathbb{R}\) is a \(2kT\)-periodic function such that
where \(\varepsilon_{0}\in(0,T)\) is a constant independent of k. Obviously, for each \(k\in\mathbb{N}\), from (1.6) we observe that \(e_{k}\in C(\mathbb{R},\mathbb{R}^{n})\) with \(e_{k}(t+2kT)\equiv e_{k}(t)\). In this paper, the approach for solving the \(2kT\)-periodic solutions to (1.5) is based on Mawhin’s continuation theorem [17], which is different from the corresponding ones in [1–4] associated to critical point theory.
The rest of this paper organized as follows. In Section 2, we state some necessary definitions and lemmas. In Section 3, we prove the main result.
2 Preliminaries
First of all, we give the definition of the homoclinic solution. A solution \(u(t)\) is named homoclinic (to 0) if \(u(t)\rightarrow0\) and \(u'(t)\rightarrow0\) as \(|t|\rightarrow+\infty\). In addition, if \(u\neq0\), then u is called a nontrivial homoclinic solution.
In the following, we recall some notations and lemmas, which are important for proving our main result.
Throughout this paper, \(\|\cdot\|\) will denote the Euclidean norm on \(\mathbb{R}^{n}\) and \(\langle\cdot,\cdot\rangle: \mathbb{R}^{n}\times\mathbb{R}^{n}\rightarrow\mathbb{R}\) denote the standard inner product.
For each \(k\in\mathbb{N}\), define
and
If the norm of \(C_{2kT}\), \(C^{1}_{2kT}\), and \(C^{2}_{2kT}\) is defined by \(\|\cdot\|_{C_{2kT}}=\|\cdot\|_{0}\), \(\|x\|_{C^{1}_{2kT}}=\max\{\| x\|_{0}, \| x'\|_{0}\}\), and \(\|x\|_{C^{2}_{2kT}}=\max\{\| x\| _{0}, \| x'\|_{0}, \| x''\|_{0}\}\), respectively, then \(C_{2kT}\), \(C^{1}_{2kT}\), and \(C^{2}_{2kT}\) are all Banach spaces.
Moreover, for any \(\psi\in C_{2kT}\), define \(\|\psi \|_{r}=( \int_{-kT}^{kT}|\psi(t)|^{r}\, dt)^{\frac{1}{r}}\), where \(r\in(1,+\infty)\).
In order to use Mawhin’s continuation theorem, we first recall it.
Let X and Y be two Banach spaces, a linear operator \(L: D(L)\subset X \rightarrow Y\) is said to be a Fredholm operator of index zero provided that
-
(a)
ImL is a closed subset of Y,
-
(b)
\(\dim\ker L=\operatorname{codim} \operatorname{Im}L<\infty\).
Let \(\Omega\subset X\) be an open and bounded set, and let \(L: D(L)\subset X \rightarrow Y\) be a Fredholm operator of index zero. This means that there are continuous linear projectors \(P: X\rightarrow X\) and \(Q: Y\rightarrow Y\) such that \(\operatorname{Im}P =\ker L\), \(\ker Q=\operatorname{Im}L\), \(X=\ker L\oplus\ker P\) and \(Y=\operatorname{Im} L\oplus \operatorname{Im} Q\). Obviously, \(L: {D}(L)\cap\ker P\rightarrow\operatorname{Im} L\) has its right inverse. Let \(K_{P}: \operatorname{Im} L\rightarrow D(L)\cap\ker P\) be the right inverse of \(L: D(L)\cap\ker P\rightarrow\operatorname{Im} L\). A continuous operator \(N:\Omega\subset X\rightarrow Y\) is said to be L-compact in \(\overline{\Omega}\) provided that
-
(c)
\(K_{p}(I-Q)N(\overline{\Omega})\) is a relative compact set of X,
-
(d)
\(QN(\overline{\Omega})\) is a bounded set of Y.
Lemma 2.1
([17])
Let X and Y be two real Banach spaces, Ω be an open and bounded subset of X, \(L: D(L)\subset X \rightarrow Y\) be a Fredholm operator of index zero and the operator \(N:\overline{\Omega}\subset X\rightarrow Y\) be L-compact in \(\overline{\Omega}\). In addition, if the following conditions hold:
- (h1):
-
\(Lx\neq\lambda Nx\), \(\forall(x,\lambda)\in\partial\Omega \times(0,1)\);
- (h2):
-
\(QNx\neq0\), \(\forall x\in\ker L\cap\partial\Omega\);
- (h3):
-
\(\deg\{ JQN,\Omega\cap\ker L,0\}\neq0\), where \(J: \operatorname{ Im}Q\rightarrow\ker L\) is a homeomorphism.
Then \(Lx=Nx\) has at least one solution in \(D(L)\cap\overline{\Omega}\).
Lemma 2.2
([18])
Let \(0<\alpha<T\) be a constant, \(\tau\in C(\mathbb{R},\mathbb{R})\) be a T-periodic function and \(\max_{t\in[0,T]}|\tau(t)|=\alpha\), then for all \(u\in C^{1}(\mathbb{R},\mathbb{R})\) with \(u(t+T)\equiv u(t)\), we have
Lemma 2.3
([3])
If \(u : \mathbb{R}\rightarrow\mathbb{R}\) is continuously differentiable on \(\mathbb{R}\), \(a>0\), \(\mu>1\), and \(p>1\) are constants, then for every \(t\in\mathbb{R}\), the following inequality holds:
Lemma 2.4
([4])
Suppose \(\tau\in C^{1}( {\mathbb{R},\mathbb {R}})\) with \(\tau(t+\omega)\equiv\tau(t)\) and \(\tau'(t)<1\), \(\forall t\in[0,\omega]\). Then the function \(t-\tau(t)\) has an inverse \(\mu(t)\) satisfying \(\mu\in C( {\mathbb{R},\mathbb{R}})\) with \(\mu(t+\omega)\equiv\mu(t)+\omega\), \(\forall t\in[0,\omega ]\).
Throughout this paper, besides τ being a periodic function with period T, we suppose in addition that \(\tau\in C^{1}( {\mathbb {R},\mathbb{R}})\) with \(\tau'(t)<1\), \(\forall t\in[0,T]\).
Remark 2.1
From the above assumption, one can find from Lemma 2.4 that the function \((t-\tau(t))\) has an inverse denoted by \(\mu(t)\). Define \(\sigma_{0}=-\min_{t\in[0,T]}\tau'(t)\), \(\sigma _{1}=\max_{t\in[0,T]}\tau'(t)\) and \(\|\tau\| _{0}=\max_{t\in[0,T]}|\tau(t)|\). Clearly, \(\sigma_{0}\geq0\) and \(0\leq\sigma_{1}<1\).
Lemma 2.5
([3])
Let \(u_{k}\in C^{2}_{2kT}\) be a \(2kT\)-periodic function for each \(k\in\mathbb{N}\) with
where \(A_{0}\), \(A_{1}\), and \(A_{2}\) are constants independent of \(k\in \mathbb{N}\). Then there exists a function \(u\in C^{1}(\mathbb {R},\mathbb{R}^{n})\) such that for each interval \([c,d]\subset\mathbb {R}\), there is a subsequence \(\{u_{k_{j}}\}\) of \(\{u_{k}\}_{k\in\mathbb {N}}\) with \(u'_{k_{j}}(t)\rightarrow u'_{0}(t)\) uniformly on \([c,d]\).
Equation (1.5) is equivalent to the following system:
where \(\varphi_{q}(s)=|s|^{q-2}s\), \(\frac{1}{p}+\frac{1}{q}=1\), \(v(t)= \varphi_{p}(\frac{u'(t)}{\sqrt{1+|u'(t)|^{2}}})=\phi^{-1}(u'(t))\).
Define
and the norm \(\|\omega\|_{X_{k}}=\|\omega\|_{Y_{k}}=\max\{\| u\|_{2kT},\| v\|_{2kT}\}\). Obviously, \(X_{k}\) and \(Y_{k}\) are Banach spaces.
Now we define the operator
where \(D(L)=\{\omega|\omega=(u(t),v(t))^{\top}: u\in C_{2kT}^{1}, u\in C_{2kT}^{1}\}\). Let
where \(B_{k}=\{x\in\mathbb{R}^{n}:|x|<1\}\). The nonlinear operator
is defined as
where Ω is an open bounded subset of \(Z_{k}\). Clearly, the problem of the existence of a \(2kT\)-periodic solution to (2.1) is equivalent to the problem of the existence of a solution in \(\overline {\Omega}\) for the equation \(L\omega=N\omega\).
By simple calculating, we have \(\ker L =\mathbb{R}^{2n}\) and \(\operatorname{Im}L =\{z\in Y_{k}, \int_{0}^{2kT}z(s)\, ds=0\}\). Therefore, L is a Fredholm operator of index zero.
Define
and
If we define \({K}_{p}=L|^{-1}_{\operatorname{Ker}L\cap D(L)}\), then it is easy to see that
where
For all \(\overline{\Omega}\) such that \(\overline{\Omega}\subset Z_{k}\subset X_{k}\), we can see that \(K_{p}(I-Q)N (\overline{\Omega})\) is a relative compact set of \(X_{k}\) and \({QN}(\overline{\Omega})\) is a bounded set of \(Y_{k}\), so the operator N is L-compact in \(\overline{\Omega}\).
For the sake of convenience, we list the following assumptions:
- (H1):
-
There are two constants \(m_{0}>0\) and \(m_{1}>0\) such that
$$\bigl\langle x,F(t,x)\bigr\rangle \leq-m_{0}|x|^{2}\quad \mbox{and}\quad \bigl\vert F(t,x)\bigr\vert \leq m_{1}|x|,\quad \mbox{for all } (t,x)\in\mathbb{R}\times\mathbb{R}^{n}. $$ - (H2):
-
There are two constants \(\alpha>0\) and \(\beta>0\) such that
$$\bigl\langle x,G(t,x)\bigr\rangle \leq-\alpha|x|^{2} \quad \mbox{and} \quad \bigl\vert G(t,x) \bigr\vert \leq \beta|x|,\quad \mbox{for all } (t,x) \in\mathbb{R}\times\mathbb{R}^{n}. $$ - (H3):
-
\(e\in C(\mathbb{R},\mathbb{R}^{n})\) is a bounded function with \(e(t)\neq0=(0,0,\ldots,0)^{\top}\) and
$$A:= \biggl(\int_{R}\bigl\vert e(t)\bigr\vert ^{2}\, dt \biggr)^{\frac{1}{2}}+\sup_{t\in R}\bigl\vert e(t)\bigr\vert < +\infty. $$
Remark 2.2
From (1.6), we can see that \(|e_{k}(t)|\leq\sup_{t\in R}|e(t)|\). So if (H3) holds, for each \(k\in\mathbb{N}\), \(( \int_{ -kT}^{kT} |e(t)|^{2}\, dt)^{\frac{1}{2}}< A\).
3 Main results
In order to study the existence of \(2kT\)-periodic solutions to system (2.1), we firstly study some properties of all possible \(2kT\)-periodic solutions to the following system:
where \((u_{k}, v_{k})^{\top}\in Z_{k}\subset X_{k}\). For each \(k\in\mathbb{N}\) and all \(\lambda\in(0,1]\). Let
This means that Δ represents the set of all the possible \(2kT\)-periodic solutions to (3.1).
Theorem 3.1
Assume that assumptions (H1)-(H3) hold, \(\frac{\alpha}{1+\sigma_{0}}> \frac{m_{1}\beta\sqrt{1-\sigma_{1}}+\sqrt {2}\beta^{2}\|\tau\|_{0}}{m_{0}(1-\sigma_{1})}\), and
where
and
Then, for each \(k\in\mathbb{N}\), if \((u, v)^{\top}\in\Delta\), there are positive constants \(\rho_{1}\), \(\rho_{2}\), \(\rho_{3}\), \(\rho_{4}\), \(A_{1}\), \(A_{2}\), \(A_{3}\), and \(A_{4}\), which are independent of k and λ, such that
Proof
For each \(k \in\mathbb{N}\), if \((u, v)^{\top}\in\Delta\), then \((u(t),v(t))^{\top}\) satisfies (3.1). Multiplying the second equation of (3.1) by \(u(t)\) and integrating from \(-kT\) to kT, we have
which combining with (H1) and (H2) gives
Furthermore,
It follows from Lemma 2.4 that
By Remark 2.1, we have
Substituting (3.3) into (3.2) and combining with \(\frac {|v(t)|^{q}}{\sqrt{1-|\varphi_{q}(v(t))|^{2}}}>|v(t)|^{q}\), we get
By applying Lemma 2.2 and (3.3), we see that
i.e.,
This implies that
and
Multiplying the second equation of (3.1) by \(u'(t)\) and integrating from \(-kT\) to kT, we have
Combining with (H1), (H2), and (3.3), we get
which results in
Substituting (3.6) into (3.5), we obtain
It follows from \(\frac{\alpha}{1+\sigma_{0}}> \frac{m_{1}\beta\sqrt {1-\sigma_{1}}+\sqrt{2}\beta^{2}\|\tau\| _{0}}{m_{0}(1-\sigma_{1})}\) that
Substituting (3.7) into (3.6), we get
i.e.,
Substituting (3.7), (3.8), and (3.9) into (3.4), we have
Moreover, it follows from Lemma 2.3 that
which combining with (3.7) and (3.9) yields
and then
Clearly, \(\rho_{1}\) is independent of k and λ.
Multiplying the second equation of (3.1) by \(v'(t)\) and integrating from \(-kT\) to kT, in view of (H1) and (H2), we have
By applying the Hölder inequality and (3.3), we have
By (3.7), (3.8), and (3.9), we have
By applying Lemma 2.3 again and combining with (3.10) and (3.12), we get
Since
we have
Clearly, \(\rho_{2}\) is independent of k and λ.
Furthermore, it follows from (3.1) that
Clearly, \(\rho_{3}\) is independent of k and λ.
Define \(F_{\rho_{3}}=\max_{|x|\leq\rho_{3},t\in[0,T]}|F(t,x)|\) and \(G_{\rho_{1}}=\max_{|y|\leq\rho_{1},t\in[0,T]}|G(t,y)|\), then from the second equation of (3.1), we get
\(\rho_{4}\) is also independent of k and λ. Therefore, from (3.7), (3.9), (3.10), (3.11), (3.12), (3.13), (3.14), and (3.15), we know that all the conclusions of Theorem 3.1 hold. □
Theorem 3.2
Assume that the conditions of Theorem 3.1 are satisfied. Then, for each \(k\in\mathbb{N}\), system (3.1) has at least one \(2kT\)-periodic solution \((u_{k}(t), v_{k}(t))^{\top}\) in \(\Delta\subset X_{k}\) such that
where \(\rho_{1}\), \(\rho_{2}\), \(\rho_{3}\), \(\rho_{4}\), \(A_{1}\), \(A_{2}\), \(A_{3}\), and \(A_{4}\) are constants defined by Theorem 3.1.
Proof
In order to use Lemma 2.1, for each \(k \in\mathbb{N}\), we consider the following system:
where \(v(t)=\varphi_{p}(\frac{\frac{u'(t)}{\lambda}}{\sqrt {1+|\frac{u'(t)}{\lambda}|^{2}}})\). Let \(\Omega_{1}\subset X_{k}\) represents the set of all possible \(2kT\)-periodic solutions of (3.16). Since \((0,1)\subset(0, 1]\), then \(\Omega_{1}\subset\Delta\), where Δ is defined by Theorem 3.1. If \((u,v)^{\top}\in\Omega_{1}\), by using Theorem 3.1, we have
Define \(\Omega_{2}=\{\omega=(u,v)^{\top}\in \ker L,QN \omega=0\}\). If \((u,v)^{\top}\in\Omega_{2}\), then \((u,v)^{\top}=(a_{1},a_{2})^{\top}\in \mathbb{R}^{2}\) (constant vector) such that
i.e.,
Multiplying the second equation of (3.17) by \(a_{1}\) and combining with (H1) and (H2), we have
Thus,
Now, if we define \(\Omega= \{\omega=(u, v)^{\top}\in X_{k},\|u\|_{0}< \rho _{1}+ \varpi, \|v\|_{0} < \frac{1+\rho_{2}}{2}<1\}\), it is easy to see that \(\Omega_{1}\cup\Omega_{2}\subset\Omega\). So, condition (h1) and condition (h2) of Lemma 2.1 are satisfied. In order to verify the condition (h3) of Lemma 2.1, define
where \(J : \operatorname{Im}Q\rightarrow\ker L \) is a linear isomorphism, \(J (u, v)=(v, u)^{\top}\). From assumption (H1), we have
Hence,
Thus, the condition (h3) of Lemma 2.1 is also satisfied. Therefore, by using Lemma 2.1, we can see that (2.1) has a \(2kT\)-periodic solution \((u_{k}, v_{k})^{\top}\in\overline{\Omega}\). Clearly, \(u_{k}\) is a \(2kT\)-periodic solution to (1.5), and \((u_{k}, v_{k})^{\top}\) must be in Δ for the case of \(\lambda= 1\). Thus, by using Theorem 3.1, we have
Hence, all the conclusions of Theorem 3.2 hold. □
Theorem 3.3
Suppose that the conditions in Theorem 3.1 hold, then (1.4) has a nontrivial homoclinic solution.
Proof
From Theorem 3.2, we see that for each \(k\in\mathbb{N}\), there exists a \(2kT\)-periodic solution \((u_{k},v_{k})^{\top}\) to (2.1) with \((u_{k},v_{k})^{\top}\in X_{k}\) and
where \(\rho_{1}\), \(\rho_{2}\), \(\rho_{3}\), \(\rho_{4}\) are constants independent of \(k \in\mathbb{N}\). Equation (3.18) together with Lemma 2.5 shows that there are a function \(w_{0}:=(u_{0},u_{0})^{\top}\in C(\mathbb {R},\mathbb{R}^{2n})\) and a subsequence \(\{(u_{k_{j}},v_{k_{j}})^{\top}\}\) of \(\{(u_{k},v_{k})^{\top}\}_{k\in\mathbb{N}}\) such that for each interval \([a,b]\subset\mathbb{R}\), \(u_{k_{j}}(t)\rightarrow u_{0}(t)\), and \(v_{k_{j}}(t)\rightarrow v_{0}(t)\) uniformly on \([a,b]\). Below, we will show that \((u_{0}(t),v_{0}(t))^{\top}\) is just a homoclinic solution to (1.4).
Since \((u_{k}(t),v_{k}(t))^{\top}\) is a \(2kT\)-periodic solution of (2.1), it follows that
For all \(a,b\in R\) with \(a < b \), there must be a positive integer \(j_{0}\) such that for \(j>j_{0}\), \([-k_{j}T,k_{j}T-\varepsilon_{0}] \supset[a-\|\tau\|_{0}, b+\|\tau\|_{0}]\). So for \(j>j_{0}\), from (1.6) and (3.19) we see that
which results in
and
uniformly for \(t\in[a,b]\) as \(j\rightarrow+\infty\). Since \(u_{k_{j}}(t)\rightarrow u_{0}(t)\) and \(u_{k_{j}}(t)\) is continuously differentiable for \(t\in(a,b)\), it follows that
which together with (3.20) yields
Similarly, by (3.21) we have
Considering a, b to be two arbitrary constants with \(a< b\), it is easy to see that \((u_{0}(t),v_{0}(t))^{\top}\), \(t\in R\), is a solution to the following equation:
i.e.,
Now, we will prove \(u_{0}(t)\rightarrow0\) and \(u_{0}'(t)\rightarrow0\) as \(|t|\rightarrow+\infty\).
Since
By using the conclusion of Theorem 3.2, we have
Let \(i\rightarrow+\infty\) and \(j \rightarrow+\infty\), we have
and then
as \(r\rightarrow+\infty\). So by using Lemma 2.3, we obtain
which implies that
Similarly, we can prove that
which together with the first equation of (3.22) gives
It is easy to see from (3.22) that \(u_{0}(t)\) is a solution for (1.4). Thus, by (3.23) and (3.24), \(u_{0}(t)\) is just a homoclinic solution to (1.4). Clearly, \(u_{0}(t)\not\equiv0\), otherwise, \(e(t)\equiv0\), which contradicts assumption (H3). Hence, the conclusion of Theorem 3.3 holds. □
Remark 3.1
Obviously, the prescribed mean curvature equations studied in [11, 12, 15, 16] are special cases of (1.4). This implies that the main result in this paper is essentially new.
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Acknowledgements
The authors express their thanks to the referee for his (or her) valuable suggestions. The research was supported by the National Natural Science Foundation of China (Grant No. 11271197).
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Lu, S., Kong, F. Homoclinic solutions for n-dimensional prescribed mean curvature p-Laplacian equations. Bound Value Probl 2015, 105 (2015). https://doi.org/10.1186/s13661-015-0362-0
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DOI: https://doi.org/10.1186/s13661-015-0362-0