Schrödinger equations with singular potentials: linear and nonlinear boundary value problems

2018/03/12 by Marcus, Moshe, Nguyen, Phuoc-Tai
#35J10 #35J60 #35J66 #35J75 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1803.04214

Abstract

Let Ω⊂ \mathbb R^N (N ≥ 3) be a C² bounded domain and F ⊂ ∂ Ω be a C² submanifold of dimension 0 ≤ k ≤ N-2. Put δ_F(x)=dist(x,F), V=δ_F^-2 in Ω and L_γV=Δ+ γV. Denote by C_H(V) the Hardy constant relative to V in Ω. We study positive solutions of equations (LE) -L_γV u = 0 and (NE) -L_γV u+ f(u) = 0 in Ω when γ< C_H(V) and f ∈ C(\mathbb R) is an odd, monotone increasing function. We establish the existence of a normalized boundary trace for positive solutions of (LE) - first studied by Marcus and Nguyen for the case F=∂ Ω - and employ it to investigate the behavior of subsolutions and super solutions of (LE) at the boundary. Using these results we study boundary value problems for (NE) and derive a-priori estimates. Finally we discuss subcriticality of (NE) at boundary points of Ω and establish existence and stability results when the data is concentrated on the set of subcritical points.

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