Siegfried Carl, Noncoercive elliptic bilateral variational inequalities in the homogeneous Sobolev space $D^{1,p}(R^N)$
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DOI: 10.23952/jnva.8.2024.4.02
Volume 8, Issue 4, 1 August 2024, Pages 503-516
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.