Measure of noncompactness, surjectivity of gradient operators and an application to the p-Laplacian

Abstract It is shown that if X is a real Banach space with dual X ⁎ and F : X → X ⁎ is a continuous gradient operator that is coercive in a certain sense and proper on closed bounded sets, then it is surjective. Use of the notion of measure of noncompactness enables sufficient conditions for properness to be given. These give rise to a surjectivity theorem for compact perturbations of strongly monotone maps and also facilitate discussion of a Dirichlet boundary-value problem involving the p -Laplacian.