Overdetermined fractional Serrin problem

Abstract In this work, we establish a Serrin-type symmetry result for a class of degenerate elliptic operators that naturally arise in the theory of generalized axially symmetric potentials. The relevant differential operator is <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>L</m:mi> <m:mi>a</m:mi> </m:msub> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:msub> <m:mo>∂</m:mo> <m:mrow> <m:mi>r</m:mi> <m:mo>⁢</m:mo> <m:mi>r</m:mi> </m:mrow> </m:msub> <m:mo>⁡</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mfrac> <m:mi>a</m:mi> <m:mi>r</m:mi> </m:mfrac> <m:mo>⁢</m:mo> <m:mrow> <m:msub> <m:mo>∂</m:mo> <m:mi>r</m:mi> </m:msub> <m:mo>⁡</m:mo> <m:mi>u</m:mi> </m:mrow> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:msub> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>y</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> </m:mrow> </m:mrow> </m:math> {L_{a}=\partial_{rr}u+\frac{a}{r}\partial_{r}u+\Delta_{y}u} , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>a</m:mi> <m:mo>≥</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {a\geq 0} represents the fractional parameter.