论文标题
根据参数，具有一类功能的规定能量的临界点：存在，多重性和分叉结果
Critical points with prescribed energy for a class of functionals depending on a parameter: existence, multiplicity and bifurcation results
论文作者
论文摘要
我们寻找具有规定能量的关键点，即均匀功能的家族$φ_μ= i_1-μi_2$，其中$ i_1，i_2 $是Banach Space $ X $上的$ C^1 $函数，而$μ\ in \ Mathbb {r} $。对于几类$φ_μ$，我们证明存在无限的许多夫妻$（μ_{n，c}，u_ {n，c}）$，因此$$φ'__ {μ_{μ_{n，c}}}}}}}（\ pm u_ _ {n，c} φ_{μ_{n，c}}（\ pm u_ {n，c}）= c \ quad \ forall n \ in \ mathbb {n}。$ $$，我们更一般地分析了问题$$φ_μ'（u）= 0，\ quad quad $ c $ c $ c $ c $ c $ c $ c $ c $ c $ c $ c $ c $ c $ c。特别是，我们表明地图$ c \mapstoμ__{n，c} $是连续的，这导致了这个问题的{\ it Energy Curves}家族。对这些曲线的分析为我们提供了几种分叉和多样性类型的结果，然后将其应用于某些椭圆问题。我们的方法基于\ cite {i1}开发的{\ it非线性通用雷利商}方法。
We look for critical points with prescribed energy for the family of even functionals $Φ_μ=I_1-μI_2$, where $I_1,I_2$ are $C^1$ functionals on a Banach space $X$, and $μ\in \mathbb{R}$. For several classes of $Φ_μ$ we prove the existence of infinitely many couples $(μ_{n,c}, u_{n,c})$ such that $$Φ'_{μ_{n,c}}(\pm u_{n,c}) = 0 \quad \mbox{and} \quad Φ_{μ_{n,c}}( \pm u_{n,c}) = c \quad \forall n \in \mathbb{N}.$$ More generally, we analyze the structure of the solution set of the problem $$Φ_μ'(u)=0, \quad Φ_μ(u)=c$$ with respect to $μ$ and $c$. In particular, we show that the maps $c \mapsto μ_{n,c}$ 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}.
