论文标题
关于非线性steklov问题的全球分叉
On global bifurcation for the nonlinear Steklov problems
论文作者
论文摘要
对于$ p \ in（1，\ infty），对于整数$ n \ geq 2 $，对于有界的lipschitz域$ω$，我们考虑以下非线性steklov分叉问题\ begin \ begin {equient {qore {qore {equient*} \ begin \ text {in} \ω，\\ | \ nabla ϕ |^{p-2} \ frac {\ partial ϕ}} {\partialν}＆=λ\ left（g | | |^|^{p-2} ϕ+ f r（ϕ+ f r（ϕ）\ right） \text{on} \ \partial Ω, \end{aligned} \end{equation*} where $Δ_p$ is the $p$-Laplace operator, $g,f \in L^1(\partial Ω)$ are indefinite weight functions and $r \in C(\mathbb R)$ satisfies $r(0)=0$ and certain growth conditions near zero and在无穷大。对于$ f，g $在某些适当的lorentz -zygmund空间中，我们确定了连续体的存在，该连续体从$（λ_1,0）$分叉，其中$λ_1$是以下非线性steklov eigenvalue eigenvalue问题的第一个特征， \ text {in} \ω，\\ | \ nabla ϕ |^{p-2} \ frac {\ partial ϕ}} {\ partialν}＆=λg| v | nabla =λg| nabla =λg| nabla | =λg| nabla | = | nabla | = |^{p-2} ϕ \ \ \ \ \ \ \ \ \ \ text {On} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \。 \ end {Aligned} \ end {equation*}
For $p \in (1, \infty),$ for an integer $N \geq 2$ and for a bounded Lipschitz domain $Ω$, we consider the following nonlinear Steklov bifurcation problem \begin{equation*} \begin{aligned} -Δ_p ϕ& = 0 \; \text{in} \ Ω, \\ |\nabla ϕ|^{p-2} \frac{\partial ϕ}{\partial ν} &= λ\left( g |ϕ|^{p-2}ϕ+ f r(ϕ) \right) \; \text{on} \ \partial Ω, \end{aligned} \end{equation*} where $Δ_p$ is the $p$-Laplace operator, $g,f \in L^1(\partial Ω)$ are indefinite weight functions and $r \in C(\mathbb R)$ satisfies $r(0)=0$ and certain growth conditions near zero and at infinity. For $f,g$ in some appropriate Lorentz-Zygmund spaces, we establish the existence of a continuum that bifurcates from $(λ_1,0)$, where $λ_1$ is the first eigenvalue of the following nonlinear Steklov eigenvalue problem \begin{equation*} \begin{aligned} -Δ_p ϕ& = 0 \; \text{in} \ Ω, \\ |\nabla ϕ|^{p-2} \frac{\partial ϕ}{\partial ν} &= λg |ϕ|^{p-2}ϕ\ \text{on} \ \partial Ω. \end{aligned} \end{equation*}
