Weak solutions for elliptic double-phase problems with logarithmic growth and nonlinear boundary conditions

Abstract

In this paper, we investigate a class of elliptic double-phase problems with logarithmic growth under nonlinear boundary conditions. The analysis is carried out in the framework of Musielak–Orlicz Sobolev spaces with variable exponents. Under natural growth assumptions on the nonlinear terms, we establish the existence of weak solutions. The proof relies on a Galerkin approximation method combined with Young measure techniques, which allow us to overcome the lack of strong compactness and to rigorously handle the passage to the limit in the nonlinear terms.