Homoclinic solutions for n-dimensional prescribed mean curvature p-Laplacian equations

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.

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.

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

1. (a)

   ImL is a closed subset of Y,

    
2. (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

1. (c)

   \(K_{p}(I-Q)N(\overline{\Omega})\) is a relative compact set of X,

    
2. (d)

   \(QN(\overline{\Omega})\) is a bounded set of Y.

    

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 \(L\omega=N\omega\).

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:

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).