Bubbling on Boundary Submanifolds for the Lin-Ni-Takagi Problem at Higher Critical Exponents

2011/07/27 by Manuel Del Pino, Manuel del Pino, Fethi Mahmoudi +4 · 3 citations
Mathematics · #35J20 #35J60 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics #math.AP #msc:35J20 #msc:35J60

paper · pdf · doi:10.48550/arxiv.1107.5566

68 pages

openalex publication_date 2011/07/27 · arxiv created 2013/08/20 · arxiv updated 2013/08/22 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28

Abstract

We consider the equation d²Δu - u+ u^{(n-k+2)/(n-k-2)} =0 \hboxinΩ, under zero Neumann boundary conditions, where Ω is open, smooth and bounded and d is a small positive parameter. We assume that there is a k-dimensional closed, embedded minimal submanifold K of ∂Ω, which is non-degenerate, and certain weighted average of sectional curvatures of ∂Ω is positive along K. Then we prove the existence of a sequence d=d_j→ 0 and a positive solution u_d such that d² |∇ u_d |² \rightharpoonup S, δ_K \ass d → 0 in the sense of measures, where δ_K stands for the Dirac measure supported on K and S is a positive constant.

Cited by

• A new minimax principle and application to the p-Laplace equation
• Some geometric and spectral aspects of restriction problems
• Concentration on minimal submanifolds for a Yamabe type problem

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