In this paper, we prove coercive a priori estimates of solutions of a Dirichlet-type boundaryvalue problem in a strip for a certain higher-order degenerate elliptic equation containing weighted derivatives of a special form up to the order 2m and ordinary partial derivatives up to the order 2k − 1 under the condition 2m > 2k − 1. At the boundary of the strip, Dirichlet-type conditions are imposed. A coercive a priori estimate for solutions of the problem considered in special weighted Sobolev-type spaces is obtained.