$T$-periodic solutions for a second order system with singular nonlinearity

Abstract

We consider a system of the form $$ \begin{align} & u''+au'=H_v(u,v)-h(t) \\ & v''+ bv'=H_u(u,v) - k(t), \end{align} $$ where $h,k $ are locally integrable and $T$-periodic, and $H$ is a $C^1$ function defined on $(0,\infty)\times (0,\infty)$, for which a good model is given by $$ H(u,v) = -( {1\over u^\alpha } + {1\over v^\beta } ),\quad \alpha ,\beta > 0 . $$ We state conditions under which existence of positive, $T$-periodic solutions for this system is ensured. We also study the problems of uniqueness and existence of multiple solutions in some special cases.