Conormal problem for a kind of nonlinear elliptic equations

We study the global boundedness of weak solutions to a class of nonlinear elliptic equations in bounded Lipschitz domains, subject to conormal boundary conditions. Using techniques from Morrey space theory, Adams-Maz’ya trace inequalities, and higher integrability properties of the gradient, we derive uniform a priori bounds for the solutions. A crucial step in our analysis is the construction of appropriate measures associated with the solution and the application of the Hartman–Stampacchia lemma, which enables us to control the solution up to the boundary of the domain.