A new kind of double phase elliptic inclusions with logarithmic perturbation terms II: Applications

This paper studies several special cases of a double phase elliptic inclusion problem (DPEI) that involves a nonlinear and nonhomogeneous partial differential operator with unbalanced growth and logarithmic perturbation terms, and two multivalued functions defined in the domain and its boundary. We establish existence and extremality results, focusing on the following two assumptions: the multivalued terms are formulated by the Clarke's generalized subdifferential operators of locally Lipschitz functions; the multivalued terms are generated by discontinuous multifunctions. When the appropriate multivalued functions are formulated by two interval functions, we develop a unifying method to construct the nontrivial sub- and supersolutions for the inequalities and inclusions under considerations, hence we obtain suitable existence and extremality results.