Ground States and Free Boundary Value Problems for the n-Laplacian in n Dimensional Space

✍ Scribed by Marta Garcı́a-Huidobro; Raúl Manásevich; James Serrin; Moxun Tang; Cecilia S. Yarur

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 164 KB
• Volume: 172
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1999.3532

✦ Synopsis

Using a new gradient estimate, we prove several theorems on the existence of radial ground states for the n-Laplace equation div( |{u| n&2 {u)+ f (u)=0 in R n , n>1, and the existence of positive radial solutions for the associated Dirichlet Neumann free boundary value problem in a ball. We treat exponentially subcritical, critical, and supercritical nonlinearities f (u), which also are allowed to have singularities at zero. Moreover, we show that the local behavior of f at zero affects the existence in a crucial way: this allows us to prove the existence of ground states for a large class of functions f (u) without imposing any restriction on their growth for large u.