My question is about examples of maximal monotone operators that are defined in non-reflexive Banach spaces and have applications in PDEs, variational inequalities, etc (any application actually)?
If possible I would like to exclude the convex subdifferential since it is a well-known example.
Take the minimizers of the energy functional $$ E(u) = \int_\Omega \frac12 |\nabla u|^2 + fu + |u| dx, \quad u\in H^1_0(\Omega). $$ The solution $u$ fulfills the variational inequality $$ \int_\Omega \nabla u \cdot (\nabla v-\nabla u) + fu + |v|-|u|dx \ge 0 \quad \forall v\in H^1_0(\Omega), $$ which can be written as a differential inclusion $$ -\Delta u + f + \lambda =0, \ \lambda \in\partial \|\cdot\|_1(u). $$ Here, the subdifferential of the $L^1$-norm is a maximal monotone operator from $L^1$ to $L^\infty$.
Another example is the following minimization problem $$ \min I_C(|\nabla u|) + \int_\Omega uf dx, $$ where $I_C$ is the indicator function of the $L^\infty$-unit ball. This leads to the infinity-Laplace equation (think $p$-Laplace with $p\to\infty$).