We study nonlinear eigenvalue problems of the type − div ( a ( x ) ∇ u ) = g ( λ , x , u ) in ℝ N , where a ( x ) is a degenerate nonnegative weight. We establish the existence of solutions and we obtain information on qualitative properties as multiplicity and location of solutions. Our approach is based on the critical point theory in Sobolev weighted spaces combined with a Caffarelli-Kohn-Nirenberg-type inequality. A specific minimax method is developed without making use of Palais-Smale condition.