Elliptic quasivariational-hemivariational inequalities in the Beppo-Levi space

In this paper, we study elliptic quasivariational-hemivariational inequalities in the Beppo–Levi space 𝐷 1 , 𝑝 ⁡ ( ℝ 𝑁 ) , where both the constraint set and the associated energy functional depend on the unknown solution. Owing to the lack of compactness of the embedding of 𝐷 1 , 𝑝 ⁡ ( ℝ 𝑁 ) into the standard Lebesgue spaces, we work within an appropriate weighted Lebesgue space framework. To overcome the difficulties arising from the solution-dependent constraints, we introduce an auxiliary penalized problem, whose solvability is established by combining a surjectivity theorem, Minty’s lemma, and Kluge’s fixed-point theorem. Finally, we establish the convergence of penalized solutions and recover a solution of the original quasivariational-hemivariational inequality.