Existence of positive solutions for nonlinear elliptic equations with convection terms (Q511409)

The authors establish the existence of positive solutions to the following nonlinear elliptic equation: \[ \begin{aligned} -\sum_{i=1}^N\frac{\partial}{\partial x_i}a_i(x,u,\nabla u) &= f(x,u,\nabla u), \quad x \in \Omega, \\ u(x) &= 0, \quad x \in \partial\Omega. \end{aligned} \] Both \((a_1,\dots,a_N)\) and \(f\) depend on \(\nabla u\). The results rely on the theory of pseudo-monotone operators, approximate solutions and truncation arguments. They rely also on a new strong maximum principle presented in this paper.