Two Classes of Nonlinear Singular Dirichlet Problems with Natural Growth: Existence and Asymptotic Behavior

Abstract

Abstract This paper is concerned with the existence, uniqueness and asymptotic behavior of classical solutions to two classes of models - △ ⁢ u ± λ ⁢ | ∇ ⁡ u | 2 u β = b ⁢ ( x ) ⁢ u - α {-\triangle u\pm\lambda\frac{|\nabla u|^{2}}{u^{\beta}}=b(x)u^{-\alpha}} , u > 0 {u>0} , x ∈ Ω {x\in\Omega} , u | ∂ ⁡ Ω = 0 {u|_{\partial\Omega}=0} , where Ω is a bounded domain with smooth boundary in ℝ N {\mathbb{R}^{N}} , λ > 0 {\lambda>0} , β > 0 {\beta>0} , α > - 1 {\alpha>-1} , and b ∈ C loc ν ⁢ ( Ω ) {b\in C^{\nu}_{\mathrm{loc}}(\Omega)} for some ν ∈ ( 0 , 1 ) {\nu\in(0,1)} , and b is positive in Ω but may be vanishing or singular on ∂ ⁡ Ω {\partial\Omega} . Our approach is largely based on nonlinear transformations and the construction of suitable sub- and super-solutions.