Positive constrained minimizers for supercritical problems in the ball

2010/06/28 by Massimo Grossi, Benedetta Noris, Grossi, Massimo +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Nonlinear Partial Differential Equations #math.AP

paper · pdf · doi:10.48550/arxiv.1006.5360

13 pages

arxiv created 2010/06/28 · openalex publication_date 2010/06/28 · arxiv updated 2010/06/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We provide a sufficient condition for the existence of a positive solution to -Δu+V(|x|) u=u^p in B₁, when p is large enough. Here B₁ is the unit ball of Rⁿ, n greater or equal to 2, and we deal both with Neumann and Dirichlet homogeneous boundary conditions. The solution turns to be a constrained minimum of the associated energy functional. As an application we show that, in case V(|x|) is smooth, nonnegative and not identically zero, and p is sufficiently large, the Neumann problem always admits a solution.

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