Noncoercive elliptic bilateral variational inequalities in the homogeneous Sobolev space $D^{1,p}(R^N)$

In this paper, we prove an existence result for a quasilinear elliptic variational inequality of the form u ∈ K ⊂ V : 0 ∈ -∆ p u+aF(ua.e. in R N }, where ∆ p is the p-Laplacian, the underlying solution space V is the homogeneous Sobolev space (also called Beppo-Levi space) V = D 1,p (R N ) with 1 < p < N, and I K : V → R ∪ {+∞} is the indicator functional corresponding to K with its subdifferential ∂ I K .The lower order Nemytskij operator F is generated by a Carathéodory function f : R N ×R → R, and the measurable and bounded coefficient a is supposed to decay like |x| -(N+α) at infinity.The growth conditions that we impose on f are such that the operator -∆ p + aF : V → V * , in general, is not coercive with respect to K which prevents us from applying standard existence results.Another difficulty, which arises due to the lack of compact embedding of V into L q (R N ) 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 V .Finally, an extension of the theory is established, which allows us to deal with noncoercive bilateral variational-hemivariational inequalities in R N .The proof of our main existence result is based on a modified penalty approach.

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