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

Abstract

Let Ω ⊂ R N (N ≥ 3) be a C 2 bounded domain andin Ω and L γV = ∆ + 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)is an odd, monotone increasing function. We establish the existence of a normalized boundary trace for positive solutions of (LE) -first studied in [17] 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.

Boundary value problems for semilinear Schrödinger equations with singular potentials and measure data

“…The first property is obvious. The second was proved in [18] and [19] for a restricted class of potentials for which precise estimates of Φ V are known. In [16,Corollary 3.5] it was shown that(1.6)(ii) holds for every potential V satisfying (A1), (A2) and two coditions on the ground state, (B1) and (B2), stated in Section 2.1.…”

“…More recently the characterization of good measures was studied with respect to the equation −L V u + f (u) = 0, mainly when V is the Hardy potential and f (t) = |t| p sign t (see, e.g. [7,10,[17][18][19]).…”

Martin kernel of Schrödinger operators with singular potentials and applications to B.V.P. for linear elliptic equations

“…these bounds were then exploited by Marcus and Nguyen [24] to derive estimates of Green and Martin kernel on layers near the boundary ∂ , which are in turn used to study respective linear and semilinear elliptic equations. Very recently, Marcus [20] has established two-sided estimates for positive L κ V -subharmonic and L κ V -superharmonic functions with V satisfying |V (x)| ≤ cd(x) −2 and provided a theory of linear equations associated to L κ V which cover several results in [23,24]. The case 0 ∈ ∂ and V (x) = |x| −2 was treated by Chen and Véron in [10] where they constructed a Poisson kernel vanishing at 0 and a singular kernel with a singularity at 0.…”

“…The case 0 ∈ ∂ and V (x) = |x| −2 was treated by Chen and Véron in [10] where they constructed a Poisson kernel vanishing at 0 and a singular kernel with a singularity at 0. Relevant works on semilinear elliptic equations involving −L κ V can be found in [9,11,17,[22][23][24][25].…”

Schrodinger equations with very singular potentials in Lipschitz domains

“…In the case of Hardy potentials, V = µ/δ 2 , estimates of L V harmonic functions in smooth domains and related results for semilinear equations have been obtained in several papers, see e.g. [9], [10], [5], [4], [8] and the references therein.…”