An improved strong comparison principle for singular $p$-Laplace equations, with applications

We consider positive weak solutions of $-{\mathrm{\Delta }}_{p}u=f(u)$, with $f$ positive and locally Lipschitz continuous. In the singular case $1<p<2$, the Harnack-type inequalities, the strong maximum principle for the linearized operator, the description of the critical set and the strong comparison principle established by Damascelli and Sciunzi [Calc.\ Var.\ Partial Differential Equations $\mathbf{\text{25}}$ (2006), 139–159] have been available, for more than twenty years, only under the restriction $\frac{2N+2}{N+2}<p<2$, a range that shrinks to the empty set as $N\to \mathrm{\infty }$. We remove the dependence on the dimension and prove all of these results for $\frac{3}{2}<p<2$. Since the strong comparison principle is used as a black box throughout the qualitative theory of $p$-Laplace equations, the improvement propagates. We show in particular that the assumption $\frac{2N+2}{N+2}<p<2$ may be replaced by $\frac{3}{2}<p<2$ in the resolution of Gibbons’ conjecture for $-{\mathrm{\Delta }}_{p}u=f(u)$ by Esposito, Farina, Montoro and Sciunzi [Math.\ Ann.\ $\mathbf{\text{382}}$ (2022), 943–974] and in their monotonicity theorem in half-spaces for changing-sign nonlinearities [Calc.\ Var.\ Partial Differential Equations $\mathbf{\text{61}}$ (2022), art.\ 154]; further applications are given in the body of the paper.