On a class of sublinear singular elliptic problems with convection term
Marius Ghergu, Vicentiu Radulescu
TL;DR
The paper analyzes a Dirichlet boundary value problem for a positive solution of $- abla^2 u + K(x) g(u) + | abla u|^a = \lambda f(x,u)$ in $\Omega$, with $u>0$ in $\Omega$ and $u=0$ on $\partial\Omega$, focusing on a singular nonlinearity $g$ and a convection term $| abla u|^a$. The authors develop an improved comparison principle and employ sub- and super-solution constructions, along with a bootstrap argument, to obtain classical solutions when $K<0$ (for all $\lambda>0$). For $K>0$, they establish a dichotomy depending on $\int_0^1 g(s) \, ds$: if this integral diverges, no classical solutions exist for any $\lambda>0$; if it converges, there exists a threshold $\lambda^*>0$ such that a classical solution exists for all $\lambda>\lambda^*$ and none for $\lambda<\lambda^*$, indicating a bifurcation with respect to $\lambda$. The results rely on conditions on $f$ (nondecreasing in the second variable and sublinear) and $g$ (nonnegative, nonincreasing) and highlight the role of Keller–Osserman-type behavior and eigenfunction-based arguments in the presence of singularity and convection.
Abstract
We establish several results related to existence, nonexistence or bifurcation of positive solutions for a Dirichlet boundary value problem with in a smooth bounded domain. The main feature of this paper consists in the presence of a singular nonlinearity, combined with a convection term. Our approach takes into account both the sign of the potential and the decay rate around the origin of the singular nonlinearity. The proofs are based on various techniques related to the maximum principle for elliptic equations.