Existence and uniqueness of solutions for a class of $p$-Laplace equations on a ball

For a class of equations generalizing the model case \Delta _p u-a(r)u^{p-1}+b(r)u^q=0 \; \; \mbox{in $B$}, \; \; u=0 \; \; \mbox{on $\partial B$}, where $B$ is the unit ball in $R^n$, $n \geq 1$, $r=|x|$, $p,q>1$, and $\Delta _p$ denotes the $p$-Laplace operator, we give conditions for the existence and uniqueness of positive solution. In case $n=1$, we give a more general result.