Quasilinear noncoercive bilateral hemivariational inequalities in $D^{1, p}(\mathbb{R}^N)$: Existence and compactness results

We consider the quasilinear elliptic hemivariational inequality in the whole $ \mathbb{R}^N $ of the form $ u\in K: \langle -{\rm{div}} \ A(x,\nabla u), v-u\rangle +\displaystyle {\int}_{ \mathbb{R}^N}a(x) j^o(x, u; v-u)\,dx\ge 0,\quad \forall v\in K, $ where $ K $ represents bilateral constraints in the homogeneous Sobolev space $ V = D^{1,p}( \mathbb{R}^N) $, also called Beppo-Levi space, given by $ K = \{v\in D^{1,p}( \mathbb{R}^N): \phi(x)\le v(x)\le \psi(x) \mbox{ for a.a. } x\in \mathbb{R}^N\}. $ It will be shown that the above hemivariational inequality is equivalent to the following multi-valued variational inequality$ u\in K: \langle -{\rm{div}} \ A(x,\nabla u)+aF(u), v-u\rangle \ge 0,\quad \forall v\in K, $where the multi-valued operator $ F $ is given by $ F(u)(x) = \partial j(x,u) $ with $ s\mapsto \partial j(x,s) $ being Clarke's generalized gradient of the locally Lipschitz function $ s\mapsto j(x,s) $. The main goal and the novelty of this paper is to prove existence and compactness results without assuming coercivity conditions on the operator $ -{\rm{div}} \ A(x,\nabla )+aF: V\to 2^{V^*} $, and without supposing the existence of sub- and supersolutions. An appropriately designed penalty technique and the use of weighted Lebesgue spaces as well as pseudomontone operator theory for multi-valued operators are the main tools in the proofs.