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We analyze the topological structure of the Nehari set for a class of functionals depending on a real parameter $\lambda$, and having two degrees of homogeneity. A special attention is paid to the extremal parameter $\lambda^*$, which is the threshold value for the Nehari set to be given by a {\it natural constraint}. The main difficulty arises when $\lambda>\lambda^*$, as the energy functional may be unbounded from below over the Nehari set. In such situation we prove the existence of local minimizers of the functional constrained to this set. We unify and extend previous existence and multiplicity results for critical points of indefinite, $(p,q)$-Laplacian, and Kirchhoff type problems.