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

Ali, Mageed; Iaia, Joseph

doi:10.32894/kujss.2024.144848.1122

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

Document Type : Research Paper

Authors

Mageed Ali ¹

Joseph Iaia ²

¹ Mathematics Department, College of Science, Kirkuk University, Kirkuk, Iraq.

² Mathematics Department, College of Science, University of North Texas, Denton, USA.

10.32894/kujss.2024.144848.1122

Abstract

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 R^{N} 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.

Keywords

exterior domains

singular

Nonlinear

existence

Subjects