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

Abstract

For the $p$-Laplace Dirichlet problem (where $\varphi (t)=t|t|^{p-2}$, $p>1$) \[ \varphi(u'(x))'+ f(u(x))=0 \;\;\;\; \mbox{for $-1<x<1$}, \;\; u(-1)=u(1)=0 \] assume that $f'(u)>(p-1)\frac{f(u)}{u}>0$ for $u>\gamma>0$, 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 $(0,\gamma)$. Uniqueness of solution follows.