Global uniqueness for the weighted $p$-Calder\'on problem

We prove that the full nonlinear Dirichlet-to-Neumann map determines every smooth positive weight in the weighted $p$-Calder\'on problem on a smooth bounded connected domain in dimension $n\ge3$, for a known exponent $1<p<\infty$, $p\ne2$. We also construct real smooth solutions with nonvanishing gradient near any prescribed compact set for arbitrary smooth positive weights in dimensions $n\ge2$ and every $1<p<\infty$. The construction couples slow amplitude and phase corrections to a Wolff wave. Its two periodic cell conditions form a planar Cauchy--Riemann system, and an adjoint estimate permits correction to an exact solution. For $p\ne2$, critical points make the linearised equation degenerate or singular. For uniqueness, it suffices to compare the nonlinear flux and the full quadratic reduced-energy form at a datum whose solution for one weight is noncritical. We derive this form from the second energy variation for a comparison background that may have interior critical points, and construct local Green kernels on its accessible regular components. A local metric continuation theorem and the nonlinear structure of the linearised tensors then give equality of the weights.