Non-singular solutions of $ p $-Laplace problems, allowing multiple changes of sign in the nonlinearity

For the $ p $-Laplace Dirichlet problem (where $ \varphi (t) = t|t|^{p-2} $, $ p > 1 $) \begin{document}$ \varphi(u'(x))'+ f(u(x)) = 0 \; \; \; \; {\rm{for}}\; -1<x<1 , \; \; u(-1) = u(1) = 0 $\end{document} assume that $ f'(u) > (p-1)\frac{f(u)}{u} > 0 $ for $ u > \gamma > 0 $, while $ \int _u^{\gamma} f(t) \, dt < 0 $ for all $ u \in (0, \gamma) $. Then any positive solution, with $ \max _{(-1, 1)} u(x) = u(0) > \gamma $, is non-singular, no matter how many times $ f(u) $ changes sign on $ (0, \gamma) $. The uniqueness of solution follows.