Solvability of the Neumann Problem in a Ball for −δu+u−v=h(|x|),v>1

✍ Scribed by Manuel A. del Pino; Gaston E. Hernandez

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 592 KB
• Volume: 124
• Category: Article
• ISSN: 0022-0396
• DOI: 10.1006/jdeq.1996.0004

✦ Synopsis

Introduction

This paper concerns mainly with the question of solvability of an elliptic problem of the form

where B is the unit ball in R N , N 2, &>1 and h # C 0 (B ) is radially symmetric. Before specifying the exact meaning we will give to a solution of (1.1) we make the following observation: If u # C 0 (B ) is a nonnegative radial function satisfying distributionally &2u+u && =h in B then u>0 on B "[0]. In fact, if u(rÄ )>0, we claim that u(r)>0 for all 0<r rÄ . To see this, we note that u satisfies the equation