Superlinear systems of second-order ODE’s

We discuss the existence of positive solutions of the system −u″=f(t,u,v,u′,v′)in (0,1),−v″=g(t,u,v,u′,v′)in (0,1),u(0)=u(1)=v(0)=v(1)=0u(0)=u(1)=v(0)=v(1)=0 where the nonlinearities ff and gg satisfy a superlinearity condition at both 0 and ∞∞. Our main result is the proof of a priori bounds for the eventual solutions. As an application, we consider the Dirichlet problem in an annulus for systems of semilinear elliptic equations with nonlinearities depending on the gradient as well. As a second application, we consider fourth-order elastic beam equations with dependence also on the derivatives u′,u″,u‴u′,u″,u‴.