2021/01/19 by Alberto Saldaña, Hugo Tavares, Saldaña, Alberto +1 · 4 citations
Computer Science · Mathematics · #35B07 #35B33 #35B38 #35B40 (Primary) #35J20 #35P30 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2101.07707
openalex publication_date 2021/01/19 · openalex created_date 2022/08/25 · openalex updated_date 2026/07/28
We study the pure Neumann Lane-Emden problem in a bounded domain -Δu = |u|p-1 u in Ω, ∂νu=0 on ∂ Ω, in the subcritical, critical, and supercritical regimes. We show existence and convergence of least-energy (nodal) solutions (l.e.n.s.). In particular, we prove that l.e.n.s. converge to a l.e.n.s. of a problem with sign nonlinearity as p\searrow 0; to a l.e.n.s. of the critical problem as p\nearrow 2^* (in particular, pure Neumann problems exhibit no blowup phenomena at the critical Sobolev exponent 2^*); and we show that the limit as p→ 1 depends on the domain. Our proofs rely on different variational characterizations of solutions including a dual approach and a nonlinear eigenvalue problem. Finally, we also provide a qualitative analysis of l.e.n.s., including symmetry, symmetry-breaking, and monotonicity results for radial solutions.