A priori upper bounds and extremal weak solutions for multi-phase problems with variable exponents

Abstract

In this paper, we deal with Dirichlet problems driven by multiphase operators with three variable exponents. As a regularity theory is still missing, we provide a priori upper bounds for the weak solutions of a very general class of such multi-phase Dirichlet problems. We emphatize that in order to achieve this result we apply De Giorgi's techniques. Then, we focus on a multi-phase Dirichlet problem with variable exponents which has in the reaction a Carathéodory function which satisfies general growth and coercivity structure conditions. We investigate the existence for such problem both of positive and negative weak solutions. Further, we produce extremal constant sign weak solutions, that is, we show that the problem under consideration admits a smallest positive weak solution and a largest negative weak solution.