Inverse problems for elliptic equations with fractional power type nonlinearities

TL;DR

This paper extends the higher order linearization method to inverse problems for semilinear elliptic equations with fractional power nonlinearities, enabling solutions even when the linear case is unknown.

Contribution

It introduces a fractional order adaptation of the higher order linearization method, broadening the applicability to general power nonlinearities.

Findings

Method successfully handles nonlinearities with fractional powers.

Results extend previous work to more general nonlinear cases.

Approach works without requiring known solutions for the linear problem.

Abstract

We study inverse problems for semilinear elliptic equations with fractional power type nonlinearities. Our arguments are based on the higher order linearization method, which helps us to solve inverse problems for certain nonlinear equations in cases where the solution for a corresponding linear equation is not known. By using a fractional order adaptation of this method, we show that the results of [LLLS20a, LLLS20b] remain valid for general power type nonlinearities.