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

TL;DR

This paper develops an abstract framework to find solutions to concave-convex problems with sign-changing or vanishing weights, providing new existence, multiplicity, and bifurcation results for associated energy functionals.

Contribution

It introduces a novel abstract method to analyze solutions of concave-convex problems with complex weight functions, extending previous approaches.

Findings

Established existence of solutions for a class of concave-convex problems.

Proved multiplicity of solutions under certain conditions.

Identified bifurcation phenomena related to parameter variations.

Abstract

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.