Existence Solutions for a Singular Nonlinear Problem with Dirichlet Boundary Conditions on Exterior Domains

This paper has proved the existence of solutions that solve the Nonlinear Partial differential equation. A study of dynamical systems has developed on the exterior of the ball centered at the origin in R N with radius R > 0 , with Dirichlet boundary conditions u = 0 on the boundary, and u ( x ) approaches 0 as | x | approaches infinity, where f ( u ) is local Lipschitzian singular at zero, and grows superlinearly as u approaches infinity. by introducing Various scalings to elucidate the singular behavior near the center and at infinity. Also, N > 2 , f ( u ) ∼ − 1 ( | u | q − 1 u for small u with 0 < q < 1 , and f ( u ) ∼ | u | p − 1 u for large | u | with p > 1 . In addition, K ( x ) ∼ | x | − α with 2 < α < 2 ( N − 1 ) for large | x | . The fixed point method and other techniques have been used to prove the existence.