On a class of sharp Sobolev type estimates with weights

We study sharp weighted Sobolev-type inequalities of the form $$\int_{0}^{1}|u(x)|\rho(x) \diff x \leqslant \Lambda \Bigl(\int_{0}^{1}|u^{(k)}(x)|^2 \diff x\Bigr)^{1/2}, \qquad u\in H_0^k(0,1),$$ where $\rho$ is a non-negative weight. We characterize the minimizers and identify the optimal constant $\Lambda(k,\rho)$ by showing that every minimizer has a constant sign and therefore solves a nonlinear eigenvalue problem of polyharmonic type. This yields an explicit characterization of extremizers for a broad class of weights. Moreover, we even provide with a an explicit computation of the optimal constant in terms of the weight function. The new weighted estimates turn to be very useful and, among other applications, allow us to recover several previous sharp estimates and Hardy type inequalities on finite intervals.