Boundary estimates for singular elliptic problems involving a gradient term

Abstract

We study bounded positive weak solutions of the singular quasilinear problem $-{\Delta }_{p}u+\vartheta |\nabla u{|}^q=u^{-\gamma }+f(u)$ in a bounded $C^2$ domain, with zero Dirichlet boundary data, where $p>1$, $\gamma >0$, $0<q\le p$, $\vartheta \ge 0$, and $f$ is continuous up to zero and locally Lipschitz on $(0,\infty )$. We establish sharp boundary asymptotics for the solution and its directional derivatives. For $\gamma >1$ the leading profile is a universal power of the boundary distance; for $\gamma =1$ it is logarithmic, with the sharp constant $(p/(p-1){)}^{1/p}$. In both cases the normalized gradient converges uniformly to the inward normal multiplied by the corresponding constant. For $0<\gamma <1$ we prove global $C^{1,\alpha }$ regularity and a classical Hopf boundary estimate. We also obtain the optimal Sobolev integrability threshold and the sharp logarithmic modulus of continuity. Finally, positivity of the total source on the solution range yields symmetry and monotonicity in symmetric domains strictly convex in one direction; when $N\ge 2$ and $\vartheta >0$, we assume $q\ge \max \{1,p/2\}$.

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