Exact multiplicity for some quasilinear elliptic Dirichlet problems where the non-linearity changes sign (Q1779845)

The author establishes the existence of one positive radially symmetric solutions \(u\) of \[ \Delta_pu + af(u) = 0 \text{ in \(B\) and \(u=0\) on \(\partial B\)}, \] satisfying \(\max u \in (b,c)\), where \(B\) is the unit ball in \(\mathbb{R}^n\) (\(n \geq 2\)), \(p > 2\), \(f\) is a continuously differentiable function such that \(f\) and \(f'\) satisfy appropriate sign conditions, and \(b, c\) are determined by \(f\). Using this fact, he shows that for \(a\) sufficiently large, the two solutions of this problem obtained by \textit{Z. Guo} and \textit{J. R. L. Webb} [J. Differ. Equ. 180, 1--50 (2002; Zbl 1014.35030)] are unique.