\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2008(2008), No. 118, pp. 1--14.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2008 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2008/118\hfil Ambrosetti-Prodi type results] {Ambrosetti-Prodi type results in a system of second and fourth-order ordinary differential equations} \author[Y. An, J. Feng \hfil EJDE-2008/118\hfilneg] {Yukun An, Jing Feng} % in alphabetical order \address{Yukun An \newline Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China} \email{anyksd@hotmail.com} \address{Jing Feng \newline Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China} \email{erma19831@sina.com} \thanks{Submitted March 17, 2008. Published August 25, 2008.} \subjclass[2000]{34B08, 34B15, 34L30, 47J30} \keywords{Differential system; Ambrosetti-Prodi type problem; subsolution; \hfill\break\indent supersolution; variational method} \begin{abstract} In this paper, by the variational method, we study the existence, nonexistence, and multiplicity of solutions of an Ambrosetti-Prodi type problem for a system of second and fourth order ordinary differential equations. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} Lazer and McKenna \cite{LM} presented the following (one-dimensional) mathematical model for the suspension bridge: \begin{equation} \label{1} \begin{gathered} y_{tt}+y_{xxxx}+\delta _1 y_t+k(y-z)^+=W(x), \quad\text{in } (0,L)\times \mathbb{R}, \\ z_{tt}-z_{xx}+\delta _2 z_t-k(y-z)^+=h(x,t), \quad\text{in } (0,L)\times \mathbb{R}, \\ y(0,t)=y(L,t)=y_{xx}(0,t)=y_{xx}(L,t)=0, \quad t\in \mathbb{R}, \\ z(0,t)=z(L,t)=0, \quad t\in \mathbb{R}. \end{gathered} \end{equation} Where the variable $z$ measures the displacement from equilibrium of the cable and the variable $y$ measures the displacement of the road bed. The constant $k$ is spring constant of the ties. When the motion of the cable is ignored, the coupled system (\ref{1}) can be simplified into a single equation which describes the motion of the road bed of suspension bridge, as follows \begin{equation} \label{2} \begin{gathered} y_{tt}+y_{xxxx}+\delta y_t+ky^+=W(x,t), \quad\text{in } (0,L)\times \mathbb{R}, \\ y(0,t)=y(L,t)=y_{xx}(0,t)=y_{xx}(L,t)=0, \quad t\in \mathbb{R}. \end{gathered} \end{equation} This Problem have been studied by many authors. In \cite{MW,CJM,CJ1}, the authors, using degree theory and the variational method, investigated the multiplicity of some symmetrical periodic solutions when $\delta =0$ and $W(x,t)=1+\epsilon h(x,t)$ or $W(x,t)=\alpha\cos x+\beta\cos 2t\cos x\epsilon $. In \cite{AZ}, the similar results for \eqref{2} are obtained in case of $\delta \neq 0$ and $W(x,t)=h(x,t)=\alpha \cos x+\beta \cos{2t}\cos{x}+\gamma \sin{2t}\cos{x}$. Those results give the conditions impose on the spring constant $k$ which guarantees the existence of multiple periodic solutions, especially the sign-changing periodic solutions in the case of $W(x,t)$ is single-sign. It is notable that the functions $\cos x, \cos{2t}\cos{x}, \sin{2t}\cos{x}$ are the eigenfunctions of linear principal operator of \eqref{2} in some function spaces. When we consider only the steady state solutions of problem (\ref{1}), we arrive at the system \begin{equation} \label{3} \begin{gathered} y_{xxxx}+k(y-z)^+=h_1(x), \quad\text{in } (0,\pi), \\ -z_{xx}-k(y-z)^+=h_2(x), \quad\text{in } (0,\pi), \\ y(0)=y(\pi)=y_{xx}(0)=y_{xx}(\pi)=0, \\ z(0)=z(\pi)=0 . \end{gathered} \end{equation} This problem has little been studied in \cite{DLT,A1}. In \cite{AF, A2}, the analogous partial differential systems have been considered when the nonlinearities $k(y-z)^+,-k(y-z)^+$ are replaced by general $f_1(y,z), f_2(y,z)$. And also, in recently, literature \cite{DM} studied the system \begin{equation} \label{4} \begin{gathered} y_{xx}+k_1y^++\epsilon z^+=\sin x, \quad\text{in } (0,\pi), \\ z_{xx}+\epsilon y^++k_2 z^+=\sin x, \quad\text{in } (0,\pi), \\ y(0)=y(\pi)=0, \\ z(0)=z(\pi)=0. \end{gathered} \end{equation} Where $u^+=\max\{u,0\}$, the constant $\epsilon$ is small enough such that the matrix $$\begin{pmatrix} k_1 & \epsilon \\ \epsilon & k_2 \end{pmatrix} $$ is a near-diagonal matrix and the positive numbers $k_1,k_2$ satisfy $$ m_1^2t_0$; \item[(ii)]has at least two solutions if $t0$ large enough, \begin{equation}\label{17} F(x,\xi)\geq 0. \end{equation} \item[(H3)] $F$ satisfies \begin{equation}\label{18} |F(x,\xi)|\leq c(|\xi_1|+|\xi_2|+1), \ \ \forall\xi\in {\mathbb{R}^2}, \ x \in (0,\pi) \end{equation} where $c>0$ is constant. \item[(H4)] For $\xi=(\xi_1,\xi_2)\in \mathbb{R}^2$ and $x\in(0,\pi)$ there holds \begin{equation}\label{9} F(x,\xi)\geq\underline{A}\xi-ce, \end{equation} for some constant $c>0$. Where $e=(1,1)$ and the matrix $\underline{A}=\begin{pmatrix} \underline{a} & \underline{b}\\ \underline{c} & \underline{d} \end{pmatrix}$ satisfies \begin{gather}\label{11} \underline{b} , \underline{c}\geq 0, \\ \label{12} (\underline{A}\xi,\xi)\leq\underline\mu|\xi|^2, \quad\text{for some }0<\underline\mu<1. \end{gather} \item[(H5)] For $\xi=(\xi_1,\xi_2)\in \mathbb{R}^2$ and $x\in(0,\pi)$ there holds \begin{equation}\label{10} F(x,\xi)\geq\overline{A}\xi-ce, \end{equation} for some constant $c>0$. Where $e=(1,1)$ and the matrix $\overline{A}=\begin{pmatrix} \overline{a} & \overline{b}\\ \overline{c} & \overline{d} \end{pmatrix}$ satisfies \begin{gather}\label{13} \overline{b} , \overline{c}\leq 0, \\ \label{14} (\overline{A}\xi,\xi)\geq\overline\mu|\xi|^2, \quad\text{for some } \overline\mu>1. \end{gather} \end{itemize} (If not mentioned, $c$ will always denote a generic positive constant.) \begin{remark} \label{rmk2.1}\rm With a simple computation it is easy to show that \eqref{11}-\eqref{12} and (\ref{13})-(\ref{14}) imply, respectively, \begin{equation}\label{15} \begin{gathered} (1-\underline{a})(1-\underline{d})-\underline{b}\underline{c}>0, \quad \underline{a},\underline{d}<1, \\ (\underline{A}-I)^{-1}\xi\leq 0, \quad \forall\xi\in \mathbb{R}^2,\; \xi\geq0, \end{gathered} \end{equation} and \begin{equation}\label{16} \begin{gathered} (1-\overline{a})(1-\overline{d})-\overline{b}\overline{c}>0, \quad \overline{a},\overline{d}>1,\\ (\overline{A}-I)^{-1}\xi\geq 0, \quad \forall\xi\in \mathbb{R}^2,\; \xi\geq0, \end{gathered} \end{equation} where $I$ is the identity matrix. \end{remark} Let $X=H_0^1(0,\pi)\times(H_0^1(0,\pi)\cap H^2(0,\pi))$ be Hilbert space with the inner product $$ \langle W,\Psi \rangle=\int_0^\pi(u'\psi_1'+v''\psi_2''), \quad \forall W=(u,v),\; \Psi=(\psi_1,\psi_2)\in X, $$ and the corresponding norm $$ \|W\|_X^2=\int_0^\pi({u'}^2+{v''}^2). $$ Consider the second-order ordinary differential eigenvalue problem \begin{gather*} -u''=\lambda u, \quad\text{in } (0,\pi), \\ u(0)=u(\pi)=0, \end{gather*} and the fourth-order ordinary differential eigenvalue problem \begin{gather*} v''''=\lambda v, \quad\text{in } (0,\pi), \\ v(0)=v(\pi)=v''(0)=v''(\pi)=0. \end{gather*} It is well known that $\lambda_1=1$ and $\varphi_1=\sin x$ are the positive first eigenvalue and the associated eigenfunction, respectively. Hence, it follows from the Poincare inequality that, for all $W\in X$, \begin{equation}\label{7} \int_0^\pi|W|^2\leq\|W\|_X^2. \end{equation} A vector $W\in X$ is a weak solution of \eqref{5} if, and only if, it is a critical point of the associated Euler-Lagrange functional \begin{equation}\label{8} J(W)={1\over2}\int_0^\pi({u'}^2+{v''}^2)-\int_0^\pi H(x,u,v)-\int_0^\pi [(t_1\sin x+h_1)u+(t_2\sin x+h_2)v] \end{equation} It is standard to show that the functional $J(W)$ is well defined, $J(W)\in C^1(X,\mathbb{R})$ and $X\to{\mathbb{R}};\ W\to \int_0^\pi H(x,u,v)+\int_0^\pi [(t_1\sin x+h_1)u+(t_2\sin x+h_2)v]$ has compact derivative under the assumptions (H1) and (H3). \begin{lemma} \label{lem2.2} Assume that {\rm (H1)-(H5)} hold. Then $J$ satisfies the (PS) condition. \end{lemma} \begin{proof} Let $\{W_n=(u_n,v_n)\}\subset X$ be a sequence such that $|J(W_n)|\leq c$ and $J'(W_n)\to 0$. This implies \begin{equation}\label{19} \begin{aligned} &\Big|\int_0^\pi(u_n'\psi_1'+v_n''\psi_2'')-\int_0^\pi\Big[ (f_1\psi_1+f_2\psi_2)+(t_1\sin x+h_1)\psi_1+(t_2\sin x+h_2)\psi_2\Big]\Big|\\ & \leq \varepsilon_n\|\Psi\|_X \end{aligned} \end{equation} for all $\Psi=(\psi_1,\psi_2)\in X$, where $\varepsilon_n\to0(n\to\infty)$. Then by the above discussion it suffices to prove that $\{W_n\}$ is bounded. \noindent\textbf{Step 1:} Show the boundedness of $\{W_n^-\}$. Let $W_n^-=(u_n^-,v_n^-)$, $w^-=\max\{0,-w\}$. Since $h_1,h_2$ are bounded, there exists $M_1,M_2\geq 0$ such that \begin{equation}\label{20} |t_1\sin x+h_1|\leq M_1, \quad |t_2\sin x+h_2|\leq M_2. \end{equation} Moreover, from \eqref{9} and \eqref{11}, we have \begin{gather*} f_1(x,u_n,v_n)(-u_n^-)\leq\underline{a}(u_n^-)^2 +\underline{b}u_n^-v_n^-+cu_n^-, \\ f_2(x,u_n,v_n)(-v_n^-)\leq\underline{d}(v_n^-)^2 +\underline{c}u_n^-v_n^-+cv_n^-. \end{gather*} Choosing $c>\max\{M_1,M_2\}$ and taking $\psi_1=-u_n^-,\psi_2=-v_n^-$ in (\ref{19}), then using the above inequalities and \eqref{12}, we obtain \begin{align*} \|W_n^-\|_X^2 & \leq \int_0^\pi(\underline{A}W_n^-,W_n^-)+ \int_0^\pi(c{u_n^-}-M_1{u_n^-}+c{v_n^-}-M_2{v_n^-})+c\|W_n^-\|_X\\ & \leq \underline{\mu}\int_0^\pi|W_n^-|^2+ d\int_0^\pi({u_n^-}+{v_n^-})+c\|W_n^-\|_X. \end{align*} Where $d\geq\max\{c-M_1,c-M_2\}$ is constant. Using H\"{o}lder inequality and Poincare inequality, we get \begin{align*} \int_0^\pi|u_n^-|\leq c(\int_0^\pi|u_n^-|^2)^{1/2}\leq c(\int_0^\pi|{u_n^-}'|^2)^{1/2}, \\ \int_0^\pi|v_n^-|\leq c(\int_0^\pi|v_n^-|^2)^{1/2}\leq c(\int_0^\pi|{v_n^-}''|^2)^{1/2}. \end{align*} Then from these two inequalities and (\ref{7}) we have $$ (1-\underline{\mu})\|W_n^-\|_X^2\leq c\|W_n^-\|_X, $$ since $0<\underline\mu<1$, $\|W_n^-\|$ is bounded. \noindent\textbf{Step 2:} Show the boundedness of $\{W_n\}$. Suppose by contradiction that $\{W_n\}$ is unbounded, then there exists a subsequence (still denote $\{W_n\}$) such that $\|W_n\|_X\to\infty \quad\text{as } n\to\infty$. Setting $V_n=(x_n,y_n)=W_n/\|W_n\|_X$, then $\|V_n\|_X=1$ and there exists a subsequence such that \begin{gather}\label{21} V_n \rightharpoonup V_0=(x_0,y_0), \quad \text{in } X, \\ \label{b1} V_n \to V_0, \quad\text{in } L^2(0,\pi)\times L^2(0,\pi), \\ V_n \to V_0, \quad\text{a.e. in } (0,\pi), \notag \\ \label{b2} \text{with } |x_n(x)|, |y_n(x)|\leq h(x)\in L^2,\; x\in (0,\pi). \end{gather} By step 1 we may assume that $ V_n^- \to 0$ in $ L^2\times L^2$ and $V_n^- \to 0$ a.e.in $ (0,\pi)$. Clearly, $V_0 \geq 0$. Denote \begin{align*} G_n(x)&=(g_n^1(x),g_n^2(x))\\ &={(f_1(x,W_n(x))+t_1\sin x+h_1, f_2(x,W_n(x))+t_2\sin x+h_2)\over\|W_n\|_X}. \end{align*} We claim that \begin{equation}\label{22} G_n\to\gamma=(\gamma_1,\gamma_2)\geq 0 \quad\text{in } L^2\times L^2. \end{equation} In fact, let $A_n=\{x\in(0,\pi);u_n(x)\leq 0 \ and \ v_n(x)\leq 0\}$ and let $\chi_n$ denotes its characteristic function, then $G_n=\chi_n G_n+(1-\chi_n)G_n$. By (H3), (\ref{b1}), (\ref{b2}) and using the Lebesgue Dominated Convergence Theorem, we get $$ \chi_n{F(x,W_n)\over\|W_n\|_X}\to 0 \quad\text{in }L^2\times L^2. $$ Moreover, from (\ref{20}) we have $$ \chi_n{(t_1\sin x+h_1,t_2\sin x+h_2)\over\|W_n\|_X}\to 0 \quad\text{in }L^2\times L^2. $$ Hence $\chi_n G_n\to 0$ in $L^2\times L^2$. With the same reasoning $(1-\chi_n)G_n\to\gamma'=(\gamma'_1,\gamma'_2)$ in $L^2\times L^2$. Therefore, we only need to prove that $\gamma'\geq0$. (i) If $u_n(x)\geq 0$ and $v_n(x)\leq 0$, since $\overline{a}>1$, from \eqref{10} we have $$ (1-\chi_n)g_n^1(x)+\overline{b}(y_n^-(x))+{c\over\|W_n\|_X}- (1-\chi_n){t_1\sin x+h_1\over\|W_n\|_X} \geq \overline{a}x_n^+(x) \geq 0 $$ and from \eqref{9} and \eqref{11}, we obtain $$ (1-\chi_n)g_n^2(x)+\underline{d}(y_n^-(x))+{c\over\|W_n\|_X}- (1-\chi_n){t_2\sin x+h_2\over\|W_n\|_X} \geq \underline{c}x_n^+(x) \geq 0 $$ Since $ V_n^- \to 0$ in $ L^2\times L^2$ and \begin{gather*} (1-\chi_n)g_n^1(x)+\overline{b}(y_n^-(x))+{c\over\|W_n\|_X}- (1-\chi_n){t_1\sin x+h_1\over\|W_n\|_X}\to\gamma'_1, \\ (1-\chi_n)g_n^2(x)+\underline{d}(y_n^-(x))+{c\over\|W_n\|_X}- (1-\chi_n){t_2\sin x+h_2\over\|W_n\|_X}\to\gamma'_2 \end{gather*} we get $\gamma'\geq0$. (ii) If $u_n(x)\leq 0$ and $v_n(x)\geq 0$, we can handle in the same way to obtain that $\gamma'\geq0$. (iii) If $u_n(x)\geq 0$ and $v_n(x)\geq 0$, the assertion $\gamma'\geq0$ can be inferred from (H2). Now dividing (\ref{19}) by $\|W_n\|_X$, using (\ref{21}), (\ref{22}) and passing to the limit we obtain \begin{equation}\label{23} \int_0^\pi (x_0'\psi_1'+y_0''\psi_2'')=\int_0^\pi (\gamma_1\psi_1+\gamma_2\psi_2), \quad \forall\Psi=(\psi_1,\psi_2)\in X. \end{equation} From \eqref{10} we have $$ {(f_1(x,W_n(x))+t_1\sin x+h_1, f_2(x,W_n(x))+t_2\sin x+h_2)\over\|W_n\|_X}\geq\overline{A}V_n-{ce\over\|W_n\|_X}. $$ Passing to the limit in this inequality we get \begin{equation}\label{24} \gamma\geq\overline{A}V_0. \end{equation} Taking $\psi_1=\sin x,\psi_2=0$ and then $\psi_1=0$, $\psi_2=\sin x$ in (\ref{23}) and using (\ref{24}), it is achieved that \begin{equation}\label{25} (\overline{A}-I) \begin{pmatrix} \int_0^\pi x_0\sin x\\ \int_0^\pi y_0\sin x \end{pmatrix} \leq 0. \end{equation} From Remark \ref{rmk2.1}, applying $(\overline{A}-I)^{-1}$ to (\ref{25}) we get $(\int_0^\pi x_0\sin x,\int_0^\pi y_0\sin x)\leq 0$. Hence $x_0=y_0=0$ a.e. So, from (\ref{23}), $\int_0^\pi(\gamma,\Psi)=0$ and taking $\Psi>0$ we have $\gamma=0$. Finally, consider $\psi_1=x_n,\psi_2=y_n$ in (\ref{19}). Dividing the resulting expression by $\|W_n\|_X$, and passing to the limit we obtain $1\leq 0$, that is impossible. \end{proof} \begin{lemma} \label{lem2.3} Suppose {\rm (H5)} hold. Then \begin{equation}\label{26} \lim_{s\to+\infty}J(s\sin x,s\sin x)=-\infty. \end{equation} \end{lemma} \begin{proof} From \eqref{10} we have \begin{gather}\label{27} H(x,u,v)\geq{\overline{a}\over2}u^2+\overline{b}uv-cu+H(x,0,v)\quad \text{as } u\geq0,\forall v, \\ \label{28} H(x,u,v)\geq{\overline{d}\over2}v^2+\overline{c}uv-cv+H(x,u,0)\quad \text{as } v\geq0,\forall u. \end{gather} Adding (\ref{27}), (\ref{28}) and using them again, \begin{align*} 2H(x,u,v) & \geq {\overline{a}\over2}u^2+(\overline{b}+\overline{c})uv+{\overline{d}\over2}v^2-cu-cv+H(x,0,v)+H(x,u,0)\\ & \geq \overline{a}u^2+(\overline{b}+\overline{c})uv+\overline{d}v^2-2cu-2cv+2H(x,0,0)\\ & \geq \overline{a}u^2+(\overline{b}+\overline{c})uv+\overline{d}v^2-2cu-2cv+2c, \quad\text{for }u,v\geq0. \end{align*} Then by (\ref{14}) we have \begin{equation}\label{29} H(x,W)\geq{\overline{\mu}\over2}|W|^2-cu-cv+c. \end{equation} Taking $W=(s\sin x,s\sin x)$, where $s>0$, from (\ref{20}) and (\ref{29}) we get \begin{align*} J(s\sin x,s\sin x) & \leq {\pi s^2\over2}(1-\overline{\mu})+ (c+M_1)\int_0^\pi s\sin x+(c+M_2)\int_0^\pi s\sin x-c\\ & \leq {\pi s^2\over2}(1-\overline{\mu})+cs-c \end{align*} since $\overline{\mu}>1$, (\ref{26}) holds. \end{proof} \section{The Ambrosetti-Prodi type result} In this section, we state and prove the Ambrosetti-Prodi type result for system \eqref{5}. We need the following concepts. \begin{definition} \label{def3.1} \rm (1) We say that a vector function $W\in X$ is a weak subsolution of \eqref{5} if $$ J'(W)(\Psi)\leq 0, \quad \forall \Psi \in X, \; \Psi\geq 0. $$ (2) $W=(u,v)\in C^2\times C^4$ is a subsolution (classical) of \eqref{5} if \begin{gather*} -u'' \leq f_1(x,u,v)+t_1\sin x+h_1, \quad\text{in } (0,\pi),\\ v''''\leq f_2(x,u,v)+t_2\sin x+h_2, \quad\text{in } (0,\pi),\\ u(0) = u(\pi)=0, \\ v(0) = v(\pi)=v''(0)=v''(\pi)=0. \end{gather*} (3) Weak supersolutions and supersolutions (classical) are defined likewise by reversing the above inequalities. \end{definition} We can easily show that each a subsolution or a supersolution of \eqref{5} is indeed also a weak subsolution or a weak supersolution, respectively. For to present the subsolution and supersolution for \eqref{5}, we firstly show a maximum principle as follows. \begin{lemma} \label{lem3.2} Let $A$ be a matrix-function with entries in $C[0,\pi]$ satisfy \eqref{11} and \eqref{12}. If $W=(u,v)\in X$ is such that \begin{equation}\label{31} \int_0^\pi(u'\psi_1'+v''\psi_2'')\geq\int_0^\pi (AW,\Psi),\quad \forall \Psi=(\psi_1, \psi_2)\in X, \end{equation} then $W\geq 0$. \end{lemma} \begin{proof} Let $\Psi=W^-=(u^-, v^-)$ in (\ref{31}), by \eqref{11} and \eqref{12}, we obtain \begin{align*} \int_0^\pi(|{u^-}'|^2+|{v^-}''|^2) &\leq\int_0^\pi(A{W^-},{W^-})-\int_0^\pi(A{W^+},{W^-})\\ &\leq\underline{\mu}\int_0^\pi|{W^-}|^2 \leq\underline{\mu}\|{W^-}\|_X^2. \end{align*} Therefore, ${W^-}=0$, i.e. $W\geq0$. \end{proof} \begin{remark} \label{3.3} \rm In the classical sense, \eqref{11} and \eqref{12} are also sufficient conditions for having a maximum principle for the problem \begin{gather*} -u'' = \underline{a}u+\underline{b}v+g_1(x), \quad\text{in } (0,\pi),\\ v''''= \underline{c}u+\underline{d}v+g_2(x), \quad\text{in } (0,\pi),\\ u(0) = u(\pi)=0, \\ v(0) = v(\pi)=v''(0)=v''(\pi)=0. \end{gather*} This is, $W=(u,v)\geq0$ if $g_1\geq0,g_2\geq0$. \end{remark} \begin{lemma} \label{lem3.4} Assume condition {\rm (H4)}, i.e. \eqref{9}, \eqref{11} and \eqref{12} hold. Then, for all $t=(t_1,t_2)\in\mathbb{R}^2$, system \eqref{5} has a subsolution $W_t$ such that, if $W^t$ is any supersolution we have \begin{equation}\label{35} W_t\leq W^t \quad\text{in } (0,\pi). \end{equation} \end{lemma} \begin{proof} We consider the system \begin{equation} \label{36} \begin{gathered} -u'' = \underline{a}u+\underline{b}v-c+t_1\sin x+h_1, \quad\text{in } (0,\pi),\\ v''''= \underline{c}u+\underline{d}v-c+t_2\sin x+h_2, \quad\text{in } (0,\pi),\\ u(0) = u(\pi)=0, \\ v(0) = v(\pi)=v''(0)=v''(\pi)=0, \end{gathered} \end{equation} where $c$ is the constant in \eqref{9} and \eqref{10}. From the hypotheses on $\underline{A}$ and $h_1,\ h_2$, (\ref{36}) has a unique solution $W_t\in C^2\times C^4$. Then, using \eqref{9} we conclude that $W_t$ is in fact a subsolution of \eqref{5}. Finally, suppose that $W^t$ is any supersolution of \eqref{5}, from \eqref{9} and applying Lemma \ref{lem3.2} directly we can get the assertion (\ref{35}). \end{proof} \begin{lemma} \label{lem3.5} Suppose {\rm (H1)} holds and $(h_1,h_2)\in C[0,\pi]\times C[0,\pi]$. Then there exists $t^0\in\mathbb{R}^2$ such that, for all $t\leq t^0$, system \eqref{5} has a supersolution $W^t$. \end{lemma} \begin{proof} Let $\overline{u},\overline{v}$ be the solution of the system \begin{equation} \label{36'} \begin{gathered} -\overline{u}'' = f_1(x,0,0)+h_1(x), \quad\text{in } (0,\pi),\\ \overline{v}''''= f_2(x,0,0)+h_2(x), \quad\text{in } (0,\pi),\\ u(0) = u(\pi)=0, \\ v(0) = v(\pi)=v''(0)=v''(\pi)=0. \end{gathered} \end{equation} Due to the locally Lipschitzian condition on $f_1,f_2$, it is possible to choose $t^0=(t_1^0,t_2^0)<0$ such that \begin{gather*} f_1(x,\overline{u},\overline{v})-f_1(x,0,0)+t_1^0\sin x\leq 0,\\ f_2(x,\overline{u},\overline{v})-f_2(x,0,0)+t_2^0\sin x\leq 0. \end{gather*} Hence, from these inequalities and the system (\ref{36'}), for all $t\leq t^0$, $W^{t^0}=(\overline{u},\overline{v})$ is a supersolution for \eqref{5}. \end{proof} \begin{lemma} \label{lem3.6} Let {\rm (H4), (H5)} hold. Then for a given $h_1,h_2$, there exists an unbounded domain $\Re$ in the plane such that if $t\in\Re$, system \eqref{5} has no supersolution. \end{lemma} \begin{proof} Suppose $W=(u,v)$ is a supersolution for \eqref{5}. Multiplying both equations of this system by $\sin x$, integration them by parts and using \eqref{9}, \eqref{10} we deduce that \begin{gather} \label{37} (\underline{A}-I) \begin{pmatrix} \rho_1\\ \rho_2 \end{pmatrix} \leq \frac{\pi}{2} \begin{pmatrix} -s_1\\ -s_2 \end{pmatrix}, \\ \label{38} (\overline{A}-I)\begin{pmatrix} \rho_1\\ \rho_2 \end{pmatrix} \leq \frac{\pi}{2}\begin{pmatrix} -s_1\\ -s_2 \end{pmatrix}. \end{gather} Where $\rho_1=\int_0^\pi u\sin x, \rho_2=\int_0^\pi v\sin x, s_1=t_1-c, s_2=t_2-c$ and $c$ is the constant in \eqref{9} and \eqref{10}. From remark \ref{rmk2.1}, applying $(\underline{A}-I)^{-1}$ and $(\overline{A}-I)^{-1}$ to (\ref{37}) and (\ref{38}), respectively, we obtain that \begin{itemize} \item[(i)] If $\rho_1\leq0$, then $s_2\leq{\underline{d}-1\over\underline{b}}s_1$ when $\underline{b}\neq0$, or $ s_1\leq0$ when $\underline{b}=0$. \item[(ii)] If $\rho_1\geq0$, then $s_2\leq{\overline{d}-1\over\overline{b}}s_1$ when $\overline{b}\neq0$, or $s_1\leq0$ when $\overline{b}=0$. \end{itemize} Therefore, independently of the sign of $\rho_1$, the pair $(s_1,s_2)$ is in a region composed of the union of two half-planes passing through the origin, each of them bounded above by a straight-line of negative or infinity slope. $\Re$ is the complement of this region in the original variables $t_1$ and $t_2$. \end{proof} Now, we are at a position to prove the Ambrosetti-Prodi type result for system \eqref{5}. \begin{theorem} \label{thm3.7} Suppose that conditions {\rm (H1)--(H5)} are satisfied and that there exists a matrix \[ A(x)=\begin{pmatrix} a(x) & b(x)\\ c(x) & d(x) \end{pmatrix}, \] with $b(x),c(x)\geq 0$ (cooperativeness condition on $A(x)$) satisfies \eqref{12} such that \begin{equation}\label{30} F(x,\xi)-F(x,\eta)\geq A(x)(\xi-\eta), \quad\text{for } \xi,\eta\in \mathbb{R}^2,\; \xi\geq\eta. \end{equation} Then there exists a continuous curve $\Gamma$ splitting $\mathbb{R}^2$ into two unbounded components $N$ and $E$ such that: \begin{itemize} \item[(1)] for each $t=(t_1,t_2)\in N$, \eqref{5} has no solution; \item[(2)] for each $t=(t_1,t_2)\in E$, \eqref{5} has at least two solutions. \end{itemize} \end{theorem} \begin{proof} For each $\theta\in \mathbb{R}$, define $$ L_\theta=\{(t_1,t_2)\in {\mathbb{R}}^2; t_2+\theta=t_1\}, $$ and $R(\theta)=\{t_1\in {\mathbb{R}}; \eqref{5}$ has a supersolution with $t\in L_\theta$ for some $t_2\in \mathbb{R}\}$. Lemmas \ref{lem3.5} and \ref{lem3.6} allows us to define the continuous curve $$ \Gamma(\theta)=(\sup R(\theta),\sup R(\theta)-\theta), $$ which splits the plane into two disjoints unbounded domains $N$ and $E$ such that for all $t\in N$ no supersolution exists for \eqref{5}, while for all $t\in E$ \eqref{5} has a supersolution. Obviously, for all $t\in N$, no solution exists for \eqref{5}, result (1) is proved. To prove result (2), now we use the abstract variational theorems to find the solutions of \eqref{5} when $t\in E$. We write \begin{align*} &\langle J'(W),\Psi \rangle\\ &= \langle W,\Psi \rangle-\int_0^\pi[(f_1(x,u,v)+t_1\sin x+h_1)\psi_1+(f_2(x,u,v)+t_2\sin x+h_2)\psi_2]. \end{align*} Given $t\in E$ there exists a supersolution $W^t=(u^t,v^t)$ and a subsolution $W_t=(u_t,v_t)$ of \eqref{5} such that $W_t\leq W^t$ in $(0,\pi)$. Let $$ M=[W_t,W^t]=\{W\in X;W_t\leq W\leq W^t\}, $$ since $W_t,W^t\in L^{\infty}$ by assumption, also $M\subset L^{\infty}$ and $H(x,W(x))+(t_1\sin x+h_1)u+(t_2\sin x+h_2)v\leq c$ for all $W\in M$ and almost every $x\in(0,\pi)$. Clearly, $M$ is a closed and convex subset of $X$, hence weakly closed. Since $M$ is essentially bounded, $J(W)\geq{1\over2}\|W\|_X^2-c$ is coercive on $M$. On the other hand, if $W_n\rightharpoonup W$ weakly in $X$, where $W_n,W\in M$, we may assume that $W_n\to W$ pointwise almost everywhere; moreover, $|H(x,W_n)+(t_1\sin x+h_1)u_n+(t_2\sin x+h_2)v_n|\leq c$ uniformly, using Lebesgue Dominated Convergence Theorem, we have \begin{align*} &\int_0^\pi H(x,W_n)+\int_0^\pi [(t_1\sin x+h_1)u_n+(t_2\sin x+h_2)v_n] \\ &\to \int_0^\pi H(x,W)+\int_0^\pi [(t_1\sin x+h_1)u+(t_2\sin x+h_2)v]. \end{align*} Hence $J$ is weakly lower semi-continuous on $M$. Then we can use \cite[Theorem 1.2]{MS} to find a vector function $W_0=(u_0,v_0)\in X$ such that $W_0\in M$ is the infimum of the functional $J$ restricted to $M$. To see that $W_0$ is a weak solution of \eqref{5}, for $\varphi=(\varphi_1,\varphi_2)\in C_0^{\infty}(0,\pi)$ and $\varepsilon>0$ let \begin{gather*} u_{\varepsilon}=\min\{u^t,\max\{u_t,u_0+\varepsilon\varphi_1\}\} =u_0+\varepsilon\varphi_1-\varphi_1^{\varepsilon}+\varphi_{1\varepsilon} \\ v_{\varepsilon}=\min\{v^t,\max\{v_t,v_0+\varepsilon\varphi_2\}\} =v_0+\varepsilon\varphi_2-\varphi_2^{\varepsilon}+\varphi_{2\varepsilon} \end{gather*} with \begin{gather*} \varphi_1^{\varepsilon}=\max\{0,u_0+\varepsilon\varphi_1-u^t\}\geq0, \\ \varphi_2^{\varepsilon}=\max\{0,v_0+\varepsilon\varphi_2-v^t\}\geq0, \end{gather*} and \begin{gather*} \varphi_{1\varepsilon}=-\min\{0,u_0+\varepsilon\varphi_1-u_t\}\geq0,\\ \varphi_{2\varepsilon}=-\min\{0,v_0+\varepsilon\varphi_2-v_t\}\geq0. \end{gather*} Note that $W_{\varepsilon}=(u_{\varepsilon},v_{\varepsilon})\in M$ and $\varphi^{\varepsilon}=(\varphi_1^{\varepsilon},\varphi_2^{\varepsilon}), \varphi_{\varepsilon}=(\varphi_{1\varepsilon},\varphi_{2\varepsilon})\in X\cap L^{\infty}(0,\pi)$. The functional $J$ is differentiable in direction $W_{\varepsilon}-W_0$. Since $W_0$ minimizes $J$ in $M$ we have $$ 0\leq\langle W_{\varepsilon}-W_0,J'(W_0)\rangle= \varepsilon\langle \varphi,J'(W_0)\rangle-\langle \varphi^{\varepsilon},J'(W_0)\rangle+ \langle \varphi_{\varepsilon},J'(W_0)\rangle, $$ so that $$ \langle \varphi,J'(W_0)\rangle \geq {1\over\varepsilon}[\langle \varphi^{\varepsilon},J'(W_0)\rangle-\langle \varphi_{\varepsilon},J'(W_0)\rangle]. $$ Now, from $W^t$ is a supersolution to \eqref{5}, we get \begin{align*} &\langle \varphi^{\varepsilon},J'(W_0)\rangle\\ &= \langle \varphi^{\varepsilon},J'(W^t)\rangle+ \langle \varphi^{\varepsilon},J'(W_0)-J'(W^t)\rangle\\ &\geq \langle \varphi^{\varepsilon},J'(W_0)-J'(W^t)\rangle\\ &= \int_{\Omega}[(u_0-u^t)'(u_0+\varepsilon\varphi_1-u^t)'+ (v_0-v^t)''(v_0+\varepsilon\varphi_2-v^t)'']\\ &\quad -\int_{\Omega}[f_1(x,W_0)-f_1(x,W^t)](u_0+\varepsilon\varphi_1-u^t)\\ &\quad -\int_{\Omega}[f_2(x,W_0)-f_2(x,W^t)](v_0+\varepsilon\varphi_2-v^t)\\ &\geq \varepsilon\int_{\Omega}[(u_0-u^t)'\varphi_1'+(v_0-v^t)''\varphi_2'']\\ &\quad -\varepsilon\int_{\Omega}|f_1(x,W_0)-f_1(x,W^t)||\varphi_1| -\varepsilon\int_{\Omega}|f_2(x,W_0)-f_2(x,W^t)||\varphi_2| \end{align*} where $\Omega=\{x\in(0,\pi);W_0(x)+\varepsilon\varphi(x)\geq W^t(x)>W_0(x)\}$. Note that meas$(\Omega)\to 0$ as $\varepsilon \to 0$. Hence by absolute continuity of the Lebesgue integral we obtain that $$ \langle \varphi^{\varepsilon},J'(W_0)\rangle \geq o(\varepsilon) $$ where $o(\varepsilon)/\varepsilon\to 0$ as $\varepsilon \to 0$. Similarly, we conclude that $\langle \varphi_{\varepsilon},J'(W_0)\rangle \leq o(\varepsilon)$; thus $$ \langle \varphi,J'(W_0)\rangle \geq 0 $$ for all $\varphi \in C_0^{\infty}(0,\pi)$. Reversing the sign of $\varphi$ and since $C_0^{\infty}(0,\pi)$ is dense in $X$ we finally get that $J'(W_0)=0$, i.e. $W_0$ is a weak solution to \eqref{5}. Then using (\ref{30}) and a Maximum Principle Lemma \ref{lem3.2}, we claim that $W_0$ is a local minimum of $J$. Suppose by contradiction that $W_0$ is not a local minimum, then for every $\varepsilon>0$ there is $\widetilde{W_\varepsilon}\in\overline{B_\varepsilon(W_0)}$ (a ball of radius $\varepsilon$ around $W_0\in X$) such that $J(\widetilde{W_\varepsilon})1$, $k_2>1$ and $\epsilon\geq0$. Then there exists a curve $\Gamma$ splitting $\mathbb{R}^2$ into two unbounded components $N$ and $E$ such that: \begin{itemize} \item[(1)] for each $t=(t_1,t_2)\in N$, \eqref{40} has no solution; \item[(2)] for each $t=(t_1,t_2)\in E$, \eqref{40} has at least two solutions. \end{itemize} \end{theorem} \begin{proof} Let \[ \overline{A}=\begin{pmatrix} k_1 & 0\\ 0 & k_2\end{pmatrix}, \quad \underline{A}=\begin{pmatrix} 0 & 0\\ 0 & 0 \end{pmatrix}. \] Then we can easily verify that the conditions of Theorem \ref{thm3.7} hold and therefore the results are follow. \end{proof} \begin{remark} \label{4.2} \rm (1) Denote by $\mu_i$ ($i=1,2$) the eigenvalues of matrix \[ A=\begin{pmatrix} k_1 & \epsilon\\ \epsilon & k_2 \end{pmatrix} \] and let $\mu_1\leq\mu_2$. It can be shown that $\mu_2>1$ since $k_1>1$ and $k_2>1$. 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