A weak maximum principle for the linearized operator of \(m\)-Laplace equations with applications to a nondegeneracy result (Q2504033)

The author considers \[ \begin{cases} -\Delta_p u= f(u)\quad &\text{in }\Omega,\\ u> 0\quad &\text{in }\Omega,\\ u= 0\quad &\text{on }\partial\Omega,\end{cases} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^d\), \(d\geq 2\), \(\Delta_p u= \text{div}(|Du|^{p-2}\,Du)\) is the \(p\)-Laplace operator, \(1< p<\infty\), and \(f\) is a locally Lipschitz continuous function. Using the Poincaré-type inequality the author proves a weak maximum principle in small domains for the linearized operator, that allows to prove a weak maximum principle for the linearized operator. The author presents for the case \(f(s)= s^q\) a nondegeneracy result in weighted Sobolev spaces.