Least energy sign-changing solution for logarithmic double phase problems with nonlinear boundary condition

In this paper we study logarithmic double phase problems with superlinear right-hand sides and nonlinear Neumann boundary condition. In particular, we show that the problem under consideration has a least energy sign-changing solution. The proof is based on the minimization of the energy functional over the related nodal Nehari manifold along with the Poincaré–Miranda existence theorem. As a result of independent interest, we prove the existence of a new and very general equivalent norm in the logarithmic Musielak–Orlicz Sobolev space. In addition, we present a priori bounds for a large class of logarithmic double phase problems involving convection terms for critical and subcritical situations.