Rigidity and symmetry breaking in a Serrin problem with a radial variable exponent

We study Serrin's overdetermined torsion problem for the $p(|x|)$-Laplacian in bounded domains containing the origin. An explicit radial solution has flux $-x/n$ and gradient modulus f(r)=(r/n)1/(p(r)-1). Comparison with this solution yields two-sided boundary-gradient bounds and a rigidity criterion: if f(r)≤ f(R₁) between the smallest and largest boundary radii R₁,R₂, then the domain is the ball BR₁(0). An associated flux identity also gives a quantitative estimate for star-shaped domains. In the opposite direction, given n≥2, k≥2, R∈(0,n), and ε>0, we construct a smooth nondecreasing radial exponent, with 0≤ p-2<ε and derivative supported in an arbitrarily thin annulus about R, for which a branch of non-spherical, strictly convex overdetermined domains bifurcates from BR(0) in harmonic degree k. The construction rests on a variational Dirichlet-to-Neumann reduction and a transversal eigenvalue crossing.