On global bifurcation for the nonlinear Steklov problems
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arXiv
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| Formato: | Preprint |
| Publicado: |
2020
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| _version_ | 1866908388089135104 |
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| author | Anoop, T. V. Biswas, Nirjan |
| author_facet | Anoop, T. V. Biswas, Nirjan |
| contents | 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*} |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2010_01622 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | On global bifurcation for the nonlinear Steklov problems Anoop, T. V. Biswas, Nirjan Analysis of PDEs 35B32, 46E30, 35J50, 35J66 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*} |
| title | On global bifurcation for the nonlinear Steklov problems |
| topic | Analysis of PDEs 35B32, 46E30, 35J50, 35J66 |
| url | https://arxiv.org/abs/2010.01622 |