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

Abstract

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.