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

For the $p$-Laplace Dirichlet problem (where $φ(t)=t|t|^{p-2}$, $p>1$) \[ φ(u'(x))'+ f(u(x))=0 \;\;\;\; \mbox{for $-1<x<1$}, \;\; u(-1)=u(1)=0 \] assume that $f'(u)>(p-1)\frac{f(u)}{u}>0$ for $u>γ>0$, while $\int_u^γf(t) \, dt < 0$ for all $u \in (0,γ)$. Then any positive solution, with $\max_{(-1,1)} u(x)=u(0)>γ$, is non-singular, no matter how many times $f(u)$ changes sign on $(0,γ)$. Uniqueness of solution follows.

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