The fractional Schrödinger equation with general nonnegative potentials. The weighted space approach

Abstract

We study the Dirichlet problem for the stationary Schrödinger fractional Laplacian equation $(-\Delta {)}^{s}u+Vu=f$ posed in bounded domain $\Omega \subset {\mathbb{R}}^n$ with zero outside conditions. We consider general nonnegative potentials $V\in L_{loc}^1(\Omega )$ and prove well-posedness of very weak solutions when the data are chosen in an optimal class of weighted integrable functions $f$. Important properties of the solutions, such as its boundary behaviour, are derived. The case of super singular potentials that blow up near the boundary is given special consideration. Related literature is commented.

References in corpus (4)

• Existence, Uniqueness and Asymptotic behaviour for fractional porous medium equations on bounded domains
• Nonlocal equations with measure data
• Sharp global estimates for local and nonlocal porous medium-type equations in bounded domains
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Cited by in corpus (3)

• The fractional Schrödinger equation with singular potential and measure data
• New applications of monotonicity methods to a class of non-monotone parabolic quasilinear sub-homogeneous problems
• Limited regularity of solutions to fractional heat and Schrödinger equations