Critical points with prescribed energy for a class of functionals depending on a parameter: existence, multiplicity and bifurcation results

TL;DR

This paper investigates the existence, multiplicity, and bifurcation of critical points with prescribed energy for a class of parameter-dependent functionals, applying a nonlinear Rayleigh quotient method to elliptic problems.

Contribution

It introduces a novel analysis of energy curves and solution structures for parameter-dependent functionals, extending bifurcation and multiplicity results.

Findings

Existence of infinitely many critical point pairs with prescribed energy.

Continuity of energy level maps with respect to parameters.

Application of results to elliptic partial differential equations.

Abstract

We look for critical points with prescribed energy for the family of even functionals $\Phi_\mu=I_1-\mu I_2$, where $I_1,I_2$ are $C^1$ functionals on a Banach space $X$, and $\mu \in \mathbb{R}$. For several classes of $\Phi_\mu$ we prove the existence of infinitely many couples $(\mu_{n,c}, u_{n,c})$ such that $$\Phi'_{\mu_{n,c}}(\pm u_{n,c}) = 0 \quad \mbox{and} \quad \Phi_{\mu_{n,c}}( \pm u_{n,c}) = c \quad \forall n \in \mathbb{N}.$$ More generally, we analyze the structure of the solution set of the problem $$\Phi_\mu'(u)=0, \quad \Phi_{\mu}(u)=c$$ with respect to $\mu$ and $c$. In particular, we show that the maps $c \mapsto \mu_{n,c}$ are continuous, which gives rise to a family of {\it energy curves} for this problem. The analysis of these curves provide us with several bifurcation and multiplicity type results, which are then applied to some elliptic problems. Our approach is based on the {\it nonlinear generalized Rayleigh quotient} method developed in \cite{I1}.