Abstract. In this paper, we prove an existence result for a quasilinear elliptic variational inequality of the form in the whole under bilateral constraints given by where is the -Laplacian, the underlying solution space is the homogeneous Sobolev space (also called Beppo-Levi space) with , and is the indicator functional corresponding to with its subdifferential . The lower order Nemytskij operator is generated by a Carathéodory function , and the measurable and bounded coefficient is supposed to decay like at infinity. The growth conditions that we impose on are such that the operator , in general, is not coercive with respect to which prevents us from applying standard existence results. Another difficulty, which arises due to the lack of compact embedding of into spaces, needs to be overcome in an appropriate way. Without assuming additional assumptions such as the existence of sub- and supersolutions, we are able not only to prove the existence of solutions, but also show the compactness of the set of all solutions in . Finally, an extension of the theory is established, which allows us to deal with noncoercive bilateral variational-hemivariational inequalities in . The proof of our main existence result is based on a modified penalty approach.
How to Cite this Article:
S. Carl, Noncoercive elliptic bilateral variational inequalities in the homogeneous Sobolev space , J. Nonlinear Var. Anal. 8 (2024), 503-516.
