Quasilinear equations with dependence on the gradient

We discuss the existence of positive solutions of the problem −(q(t)φ(u′(t)))′=f(t,u(t),u′(t)) for t∈(0,1) and u(0)=u(1)=0−(q(t)φ(u′(t)))′=f(t,u(t),u′(t)) for t∈(0,1) and u(0)=u(1)=0, where the nonlinearity ff satisfies a superlinearity condition at 0 and a local superlinearity condition at +∞+∞. This general quasilinear differential operator involves a weight qq and a main differentiable part φφ which is not necessarily a power. Due to the superlinearity of ff and its dependence on the derivative, a condition of the Bernstein–Nagumo type is assumed, also involving the differential operator. Our main result is the proof of a priori bounds for the eventual solutions. The presence of the derivative in the right-hand side of the equation requires a priori bounds not only on the solutions themselves, but also on their derivatives, which brings additional difficulties. As an application, we consider a quasilinear Dirichlet problem in an annulus{−div(A(|∇u|)∇u)=f(|x|,u,|∇u|)in r1<|x|