The fractional Schrödinger equation with general nonnegative potentials. The weighted space approach. (English) Zbl 1402.35093
Summary: We study the Dirichlet problem for the stationary Schrödinger fractional Laplacian equation \((- \varDelta)^s u + V u = f\) posed in bounded domain \(\varOmega \subset \mathbb{R}^n\) with zero outside conditions. We consider general nonnegative potentials \(V \in L_{l o c}^1(\varOmega)\) 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 since it leads to so-called flat solutions. We comment on related literature.
MSC:
| 35J10 | Schrödinger operator, Schrödinger equation |
| 35D30 | Weak solutions to PDEs |
| 35J67 | Boundary values of solutions to elliptic equations and elliptic systems |
| 35J75 | Singular elliptic equations |
| 35R11 | Fractional partial differential equations |
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