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 $$\Phi_\lambda(u)=c \quad \mbox{and} \quad \Phi'_\lambda(u)=0,$$ for some suitable values of $c \in \mathbb{R}$. Here $\Phi_\lambda$ 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 $\Phi'_\lambda(u)=0$ with $\lambda$ fixed.