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

Abstract

We provide an abstract approach to find couples $(\lambda ,u)\in \mathbb{R}\times X$ satisfying $$Φ_λ(u)=c \quad \mbox{and} \quad Φ'_λ(u)=0,$$ 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 ${\Phi }_{\lambda }^{\prime }(u)=0$ with $\lambda$ fixed.