Problems for elliptic singular equations with a gradient term. (English) Zbl 1103.35031

The author studies the Dirichlet problem \(\Delta u+g(u)| \nabla u| ^q +f(u)=0\) in \(D\), \(u=0\) on \(\partial D\), where \(D\) is a bounded smooth domain in \(\mathbb R^n\), \(0<q<2\), \(g(t)\) is a decreasing smooth function in \((0,\infty )\) and \(f(t)\) is a decreasing and positive smooth function in \((0,\infty )\), which approaches infinity as \(t\to \infty \). The existence and uniqueness of a positive classical solution of the problem is proved. The asymptotic behaviour of a solution near the boundary is studied.


35J60 Nonlinear elliptic equations
35B40 Asymptotic behavior of solutions to PDEs
35J25 Boundary value problems for second-order elliptic equations
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