In this paper we consider a class of generalized quasi-variational inequalities. The variational problem is studied in the convex set <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$X\times Y$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>X</mi> <mo>×</mo> <mi>Y</mi> </mrow> </math> </EquationSource> </InlineEquation>, with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$Y$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>Y</mi> </math> </EquationSource> </InlineEquation> bounded and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$X$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>X</mi> </math> </EquationSource> </InlineEquation> unbounded. In the latter settings, we investigate about the solvability of the problem....</equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation>