Renormalized solutions of nonlinear elliptic problems in generalized Orlicz spaces

Abstract

We study a general class of nonlinear elliptic problems associated with the differential inclusion β(x,u)−div(a(x,∇u)+F(u))∋f, where f∈L1(Ω). The vector field a(⋅,⋅) is monotone in the second variable and satisfies a non-standard growth condition described by an x-dependent convex function that generalizes both Lp(x) and classical Orlicz settings. Using truncation techniques and a generalized Minty method in the functional setting of non-reflexive spaces we prove existence of renormalized solutions for general L1-data. Under an additional strict monotonicity assumption uniqueness of the renormalized solution is established. Sufficient conditions are specified which guarantee that the renormalized solution is already a weak solution to the problem.