We provide necessary and sufficient conditions for the space of smooth functions with compact supports $$C^\infty _c(\Omega )$$ to be dense in Musielak–Orlicz spaces $$L^\Phi (\Omega where $$\Omega $$ is an open subset $${\mathbb {R}}^d$$ . In particular, we prove that if $$\Phi satisfies condition $$\Delta _2$$ , closure )\cap L^\Phi equal only measure singular points zero. This extends earlie...

Abstract The main purpose of this article is to establish some new characterizations the (variable) Lipschitz spaces in terms boundedness commutator multilinear fractional Calderón-Zygmund integral operators context variable exponent Lebesgue spaces. authors do so by applying techniques Fourier series and operator, as well pointwise estimates for commutators. key tool obtaining such a estimate ...

We study unbounded weak supersolutions of elliptic partial differential equations with generalized Orlicz (Musielak--Orlicz) growth. show that they satisfy the Harnack inequality optimal exponent provided belong to a suitable Lebesgue or Sobolev space. Furthermore, we establish sharpness our central assumptions.

We prove the existence of at least three weak solutions for the Dirichlet problem when the nonlinear term f is sublinear and p(x) is greater than n. This Dirichlet problem involves a general elliptic operator in divergence form (in particular, a p(x)-Laplace operator). Our method relies upon a recent critical point theorem obtained by Bonanno and Marano, and is combined with the theory of varia...