Schr{ö}dinger operators with Leray-Hardy potential singular on the boundary

Abstract

We study the kernel function of the operator u $\to$ L $\mu$ u = --$\Delta$u + $\mu$ |x| 2 u in a bounded smooth domain $\Omega$ $\subset$ R N + such that 0 $\in$ $\partial$$\Omega$, where $\mu$ $\ge$ -- N 2 4 is a constant. We show the existence of a Poisson kernel vanishing at 0 and a singular kernel with a singularity at 0. We prove the existence and uniqueness of weak solutions of L $\mu$ u = 0 in $\Omega$ with boundary data $\nu$ + k$\delta$ 0 , where $\nu$ is a Radon measure on $\partial$$\Omega$ \ {0}, k $\in$ R and show that this boundary data corresponds in a unique way to the boundary trace of positive solution of L $\mu$ u = 0 in $\Omega$.

References in corpus (1)

• Existence and stability of solutions of general semilinear elliptic equations with measure data

Cited by in corpus (1)

• Green kernel and Martin kernel of Schrödinger operators with singular potential and application to the B.V.P. for linear elliptic equations