We present a new approach for second-order maximum entropy (max-ent) meshfree approximants that produces positive and smooth basis functions of uniform aspect ratio even for nonuniform node sets and prescribes robustly feasible constraints for the entropy maximization program defining the approximants. We examine the performance of the proposed approximation scheme in the numerical solution by a direct Galerkin method of a number of PDEs, including structural vibrations, elliptic second-order PDEs, and fourth-order PDEs for Kirchhoff–Love thin shells and for a phase field model describing the mechanics of biomembranes. The examples highlight the ability of the method to deal with nonuniform node distributions and the high accuracy of the solutions. Surprisingly, the first-order meshfree max-ent approximants with large supports are competitive when compared with the proposed second-order approach in all the tested examples, even in the higher order PDEs. Copyright © 2012 John Wiley & Sons, Ltd.