English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We look for critical points with prescribed energy for the family of evenfunctionals $\Phi_\mu=I_1-\mu I_2$, where $I_1,I_2$ are $C^1$ functionals on aBanach space $X$, and $\mu \in \mathbb{R}$. For several classes of $\Phi_\mu$we prove the existence of infinitely many couples $(\mu_{n,c}, u_{n,c})$ suchthat $$\Phi'_{\mu_{n,c}}(\pm u_{n,c}) = 0 \quad \mbox{and} \quad\Phi_{\mu_{n,c}}( \pm u_{n,c}) = c \quad \forall n \in \mathbb{N}.$$ Moregenerally, 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 $c$. Inparticular, we show that the maps $c \mapsto \mu_{n,c}$ are continuous, whichgives rise to a family of {\it energy curves} for this problem. The analysis ofthese 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}.

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