Remarks on Surjectivity of Gradient Operators

Let X be a real Banach space with dual X∗ and suppose that F:X→X∗. We give a characterisation of the property that F is locally proper and establish its stability under compact perturbation. Modifying an recent result of ours, we prove that any gradient map that has this property and is additionally bounded, coercive and continuous is surjective. As before, the main tool for the proof is the Ekeland Variational Principle. Comparison with known surjectivity results is made; finally, as an application, we discuss a Dirichlet boundary-value problem for the p-Laplacian (1<p<∞), completing our previous result which was limited to the case p≥2.