AI 中文总结
该研究针对带非线性Goldstein-Wentzell边界条件的半线性椭圆方程双椭圆问题，证明其在势阱能级深度存在非平凡解，且在更高能级有无穷多解。
AI 中文摘要
本文研究双椭圆问题非平凡解的存在性与多重性，该问题为：$$\begin{cases} -\nabla u=f(u) & 在\boldsymbol{\text{Ω}}内,\ \boldsymbol{u}=0 & 在\boldsymbol{\text{Γ}_0}上,\ -\nabla_\boldsymbol{\text{Γ}} u +\boldsymbol{\text{∂}}_\boldsymbol{\text{ν}} u =g(u) & 在\boldsymbol{\text{Γ}_1}上, \\ \text{其中Ω是}\boldsymbol{\text{R}}^\boldsymbol{\text{N}}(\boldsymbol{\text{N≥2}})的有界开区域，具有\boldsymbol{\text{C}}^\boldsymbol{\text{1}}边界\boldsymbol{\text{Γ=∂Ω}}，且\boldsymbol{\text{Γ=Γ}_0\boldsymbol{\text{∪Γ}_1}，\boldsymbol{\text{Γ}_0\boldsymbol{\text{∩Γ}_1}=∅，\boldsymbol{\text{Γ}_1}非空且在\boldsymbol{\text{Γ}}上相对开，\boldsymbol{\text{H}}^\boldsymbol{\text{N-1}}(\boldsymbol{\text{Γ}_0})>0。项\boldsymbol{f}和\boldsymbol{g}分别在\boldsymbol{\text{Ω}}和\boldsymbol{\text{∂Ω}}上关于Sobolev嵌入是次临界的。我们证明，在适当假设下，该问题在势阱能级深度（即非平凡解的最小能级）处存在非平凡解，且在更高能级处有无穷多解。
英文摘要
The paper deals with the existence and multiplicity of nontrivial solutions for the doubly elliptic problem $$\begin{cases} -Δu=f(u) \qquad &\text{in $Ω$,}\\ \phantom{-}u=0 &\text{on $Γ_0$,}\\ -Δ_Γu +\partial_νu =g(u)\qquad &\text{on $Γ_1$,} \end{cases} $$ where $Ω$ is a bounded open domain of $\mathbb{R}^N$ ($N\ge 2$) with $C^1$ boundary $Γ=\partialΩ$, with $Γ=Γ_0\cupΓ_1$, $Γ_0\capΓ_1=\emptyset$, $Γ_1$ being nonempty and relatively open on $Γ$, $\mathcal{H}^{N-1}(Γ_0)>0$. The terms $f$ and $g$ are subcritical with respect to Sobolev embeddings, respectively in $Ω$ and on $\partialΩ$. We prove that, under suitable assumptions, the problem admits nontrivial solutions at the depth of the potential well energy level, which is the minimum energy level for nontrivial solutions. We also prove that the problem has infinitely many solutions at higher energy levels.
