On the Boundedness of Solutions to Elliptic Variational Inequalities

Abstract

Abstract. In this paper we present global a priori bounds for a class of variational inequalities involving general elliptic operators of second-order and terms of generalized directional derivatives. Based on Moser's and De Giorgi's iteration technique we prove the boundedness of solutions of such inequalities under certain criteria on the set of constraints. In our proofs we also use the localization method with a certain partition of unity and a version of a multiplicative inequality estimating the boundary integrals. Some sets of constraints satisfying the required conditions are stated as well.

Moser iteration applied to elliptic equations with critical growth on the boundary

“…Furthermore, we only prove the case when q 1 = p * and q 2 = p * . The other cases were already obtained in [21, Theorem 4.1] and [22,Theorem 3.1]. Moreover, we will denote positive constants with M i and if the constant depends on the parameter κ we write M i (κ) for i = 1, 2, .…”

“…Finally, we mention some works concerning boundedness and regularity results of weak solutions to quasilinear equations of the form (1.1) that have subcritical growth, see, for example, Fan-Zhao [3], Gasiński-Papageorgiou [5], [7, pp. 737-738], Hu-Papageorgiou [9], Lê [10], Motreanu-Motreanu-Papageorgiou [13], Pucci-Servadei [16], Winkert [20], [21], [22], Winkert-Zacher [23], [24], [25] and the references therein. The methods used in these papers are mainly based on Moser's iteration or De Giorgi's iteration technique and no critical growth occurs.…”

The boundedness and Hölder continuity of weak solutions to elliptic equations involving variable exponents and critical growth

“…Finally, we mention papers which are very close to our topic dealing with certain types of a priori bounds for equations with p-or p(•)-structure. We refer to Ding-Zhang-Zhou [10,11], García Azorero-Peral Alonso [17], Guedda-Véron [21], Marino-Winkert [32,33], Pucci-Servadei [40], Wang [43], Winkert [44,46], Winkert-Zacher [49], Zhang-Zhou [50], Zhang-Zhou-Xue [51] and the references therein.…”