A priori estimates and Liouville type results for quasilinear elliptic equations involving gradient terms (Q6622471)

This article is concerned with the study of qualitative properties of solutions to \(-\Delta_m u=|u|^{p-1}u+M|\nabla u|^q\) in a domain \(\Omega\subset \mathbb{R}^N\). Here \(m>1\), \(M\in \mathbb{R}\) and \(p, q>0\). The authors employ a direct Bernstein method to derive a-priori estimates on the gradient term of the type \(|\nabla u(x)|\leq C(M^{\gamma_1}+\mathrm{dist}(x, \partial\Omega)^{\gamma_2})\) for some \(C, \gamma_1, \gamma_2\) depending on the exponents \(p, q,m\). Various universal bounds on the solution \(u\) are deduced, which further yield Liouville type results. The results are new to the literature and expand on the existent one for quasilinear elliptic equations with nonlinear gradient terms.