Existence theorems for the Dirichlet elliptic inclusion involving exponential-growth-type multivalued right-hand side

We present two existence results for the Dirichlet elliptic inclusion with an upper semicontinuous multivalued right-hand side in exponential-type Orlicz spaces involving a vector Laplacian, subject to Dirichlet boundary conditions on a domain Ω ⊂ R2. The first result is obtained via the multivalued version of the Leray–Schauder principle together with the Nakano–Dieudonné sequential weak compactness criterion. The second result is obtained by using the nonsmooth variational technique together with a formula for Clarke's subgradient for Lipschitz integral functionals on “nonregular” Orlicz spaces.