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

Abstract

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

6 pages, 1 figure