Existence of radially symmetric solutions of the inhomogeneous $p$-Laplace equation

We consider the Dirichlet problem for the inhomogeneous $p$-Laplace equation with $p$ nonlinear source. New sufficient conditions are established for the existence of weak bounded radially symmetric solutions as well as a priori estimates of solution and of the gradient of solution. We obtain an explicit formula that shows the dependence of the existence of these solutions on the dimension of the problem, the size of the domain, the exponent $p$, the nonlinear source, and the exterior mass forces.