Existence results of variable exponent double-phase multivalued elliptic inequalities with logarithmic perturbation and convections

Abstract In this study, we deal with a multivalued elliptic variational inequality involving a logarithmic perturbed variable exponents double-phase operator. Additionally, it features a multivalued convection term alongside two multivalued terms, one defined within the domain and the other on its boundary. Under the noncoercive framework, we establish the existence results of weak solutions for the multivalued inequality by employing a surjective theorem for multivalued pseudomonotone operators along with the penalty technique. On the other hand, we prove the compactness of solution set by employing the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>S</m:mi> </m:mrow> <m:mrow> <m:mo>+</m:mo> </m:mrow> </m:msub> </m:math> {S}_{+} -property of the associated perturbed variable exponent double-phase operator. Finally, we focus on special cases to the multivalued inequality, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>K</m:mi> </m:math> K is a bilateral constraint set, and the two multivalued terms are Clarke’s generalized gradients with respect to two locally Lipschitz functions.