Prescribed energy solutions of concave-convex type problems involving sign-changing or vanishing weights

We provide an abstract approach to find couples $(\lambda ,u)\in \mathbb{R}\times X$ satisfying $${\mathrm{\Phi }}_{\lambda }(u)=cand{\mathrm{\Phi }}_{\lambda }^{\mathrm{\prime }}(u)=0,$$ for some suitable values of $c\in \mathbb{R}$. Here ${\mathrm{\Phi }}_{\lambda }$ is a $C^1$ functional (set on a Banach space $X$) whose main prototype is the energy functional associated to a concave-convex problem with sign-changing or vanishing weights. This approach allows us to derive several existence, multiplicity and bifurcation type results for the equation ${\mathrm{\Phi }}_{\lambda }^{\mathrm{\prime }}(u)=0$ with $\lambda$ fixed.