The Musielak–Orlicz setting unifies variable exponent, Orlicz, weighted Sobolev, and double-phase spaces. They inherit technical difficulties resulting from general growth and inhomogeneity. In this survey we present an overview of developments of the theory of existence of PDEs in the setting including reflexive and non-reflexive cases, as well as isotropic and anisotropic ones. Particular attention is paid to problems with data below natural duality in absence of Lavrentiev’s phenomenon.

A pocket guide to nonlinear differential equations in Musielak–Orlicz spaces

In this paper we prove the uniform boundedness of the operators of co- nvolution in the Musielak-Orlicz spaces and the density of C 1 0 (R n ) in the Musielak-

An approximation theorem in Musielak-Orlicz-Sobolev spaces

In this work, we shall be concerned with the existence and uniqueness result to the nonlinear parabolic equations whose prototype is $$\begin{aligned} \left\{ \begin{array}{lll} \displaystyle \frac{\partial b( u)}{\partial t} -\varDelta _{M}u - \text{ div } \left( \overline{c}(x,t)\overline{M}^{-1}M\left( \frac{\alpha _{0}}{\lambda }|b(u)|\right) \right) =f &{} \text {in}&{} Q_{T},\\ \displaystyle u(x,t)=0 &{} \text {on} &{} \partial \varOmega \times (0,T),\\ b(u)(t=0)=b(u_{0}) &{} \text {in} &{} \varOmega , \end{array} \right. \end{aligned}$$
 where $$ -\,\varDelta _{M}u{=}-\,\text{ div } ((1+|u|)^{2}Du\frac{\log (e+Du)}{|Du|})$$
 , $$\overline{c}\in (L^{\infty }(Q_{T}))^{N}$$
 and $$M(t)= t\log (e+t)$$
 is an N-function. The data f and $$b(u_{0})$$
 in $$L^{1}(Q_T)$$
 and $$L^{1}(\varOmega )$$
 .

Existence and uniqueness of a renormalized solution of parabolic problems in Orlicz spaces

Elliptic inequalities with lower order terms and L1 data in Orlicz spaces

/pdf/nonlinear-unilateral-problems-in-orlicz-spaces-2w3m2kw7vi.pdf

Nonlinear unilateral problems in Orlicz spaces

An existence result for a strongly non-linear elliptic equation with natural growth condition on the non-linearity and L1-data is proved.

/pdf/non-linear-elliptic-problems-having-natural-growth-and-l1-3gahbehcd6.pdf

Non-linear elliptic problems having natural growth and L1 data in Orlicz spaces

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Non-linear elliptic problems having natural growth and L1 data in Orlicz spaces