Min-max solutions for the impulsive boundary value problems for the \((p_1(t),p_2(t))\)-Laplacian systems (Q2849207)

The paper deals with the Dirichlet impulsive problem of the form NEWLINE\[NEWLINE\left\{\begin{aligned} &-\frac{d}{dt}\left(\left|\frac{d}{dt}x(t)\right|^{p_1(t)-2}\frac{d}{dt}x(t)\right)+F_x(t,x(t),y(t))+f_1(t)=0,\\ &\frac{d}{dt}\left(\left|\frac{d}{dt}y(t)\right|^{p_2(t)-2}\frac{d}{dt}y(t)\right)+F_y(t,x(t),y(t))+f_2(t)=0,\\ &x(0)=x(\pi)=0, ~~~ y(0)=y(\pi)=0,\\ &\left|\frac{d}{dt}x(t_j^+)\right|^{p_1(t_j^+)-2}\frac{d}{dt}x(t_j^+)-\left|\frac{d}{dt}x(t_j^-)\right|^{p_1(t_j^-)-2}\frac{d}{dt}x(t_j^-)=I_j(x(t_j)), ~ j=1,2, \dots, m_1,\\ &-\left|\frac{d}{dt}y(t_j^+)\right|^{p_2(t_j^+)-2}\frac{d}{dt}y(t_j^+)+\left|\frac{d}{dt}y(t_j^-)\right|^{p_1(t_j^-)-2}\frac{d}{dt}y(t_j^-)=I_j(y(t_j)), ~ j=m_1+1,\dots, m_2, \end{aligned}\right. NEWLINE\]NEWLINE where \(p_1,p_2\in C([0,\pi], \mathbb R^+),\) \(0<t_1<t_2<\dots<t_{m_{1}}<\dots<t_{m_{2}}<\pi,\) \(0<m_1<m_2\) fixed integers, \(F, F_x, F_y: [0,\pi]\times \mathbb R\times\mathbb R\to \mathbb R\) are Carathéodory functions, \(I_j: \mathbb R\to \mathbb R\) are continuous for \(j=1,\dots, m_2\) and \(f_1, f_2\in L^1(0,\pi)\) such that \(f_i(t)\neq 0\) for a.e. \(t\in [0,\pi]\), \(i=1,2\).NEWLINENEWLINEUsing Ky Fan's min-max theorem, the author proves, under some additional assumptions, the existence of at least one solution to the given problem.