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

This paper proves the existence of solutions that solve the Nonlinear Partial differential equation on the exterior of the ball centered at the origin in RN with radius R > 0, with boundary conditions u = 0 on the boundary, and u ( x ) approaches 0 as | x | approaches infinity. When the function is local Lipschitzian grows superlinear at infinity and singular at 0. 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. Also, K ( x ) ~ | x |^ { - ( Alpha) } with 2 < Alpha < 2 ( N - 1 ) for large | x |. We used the fixed point method and other techniques to prove the existence.