We study global uniqueness in an inverse problem for the fractional semilinear Schrödinger equation (-Δ)su + q(x, u) = 0 with s ε (0, 1). We show that an unknown function q(x, u) can be uniquely determined by the Cauchy data set. In particular, this result holds for any space dimension greater than or equal to 2. Moreover, we demonstrate the comparison principle and provide an L ∞ estimate for this nonlocal equation under appropriate regularity assumptions.
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Received by the editors November 13, 2017, and, in revised form, June 27, 2018. 2010 Mathematics Subject Classification. Primary 35B50, 35R30, 47J05, 65N21, 35R11. Key words and phrases. Calderón’s problem, partial data, semilinear, fractional Schrödinger equation, nonlocal, maximum principle. The second author was supported in part by MOST of Taiwan 160-2917-I-564-048.
- Calderón’s problem
- Fractional schrödinger equation
- Maximum principle
- Partial data