Elliptic $p$-Laplacian systems with nonlinear boundary condition
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909069571260416 |
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| author | Borer, Franziska Carl, Siegfried Winkert, Patrick |
| author_facet | Borer, Franziska Carl, Siegfried Winkert, Patrick |
| contents | In this paper we study quasilinear elliptic systems given by
\begin{equation*}
\begin{aligned}
-Δ_{p_1}u_1
& =-|u_1|^{p_1-2}u_1 \quad && \text{in } Ω,\newline
-Δ_{p_2}u_2
& =-|u_2|^{p_2-2}u_2 \quad && \text{in } Ω,\newline
|\nabla u_1|^{p_1-2}\nabla u_1 \cdot ν
&=g_1(x,u_1,u_2) && \text{on } \partialΩ,\newline
|\nabla u_2|^{p_2-2}\nabla u_2 \cdot ν
&=g_2(x,u_1,u_2) && \text{on } \partialΩ,
\end{aligned}
\end{equation*}
where $ν(x)$ is the outer unit normal of $Ω$ at $x \in \partialΩ$, $Δ_{p_i}$ denotes the $p_i$-Laplacian and $g_i\colon \partialΩ\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ are Carathéodory functions that satisfy general growth and structure conditions for $i=1,2$. In the first part we prove the existence of a positive minimal and a negative maximal solution based on an appropriate construction of sub- and supersolution along with a certain behavior of $g_i$ near zero related to the first eigenvalue of the $p_i$-Laplacian with Steklov boundary condition. The second part is related to the existence of a third nontrivial solution by imposing a variational structure, that is, $(g_1,g_2)=\nabla g$ with a smooth function $(s_1,s_2)\mapsto g(x,s_1,s_2)$. By using the variational characterization of the second eigenvalue of the Steklov eigenvalue problem for the $p_i$-Laplacian together with the properties of the related truncated energy functionals, which are in general nonsmooth, we show the existence of a nontrivial solution whose components lie between the components of the positive minimal and the negative maximal solution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_05846 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Elliptic $p$-Laplacian systems with nonlinear boundary condition Borer, Franziska Carl, Siegfried Winkert, Patrick Analysis of PDEs In this paper we study quasilinear elliptic systems given by \begin{equation*} \begin{aligned} -Δ_{p_1}u_1 & =-|u_1|^{p_1-2}u_1 \quad && \text{in } Ω,\newline -Δ_{p_2}u_2 & =-|u_2|^{p_2-2}u_2 \quad && \text{in } Ω,\newline |\nabla u_1|^{p_1-2}\nabla u_1 \cdot ν &=g_1(x,u_1,u_2) && \text{on } \partialΩ,\newline |\nabla u_2|^{p_2-2}\nabla u_2 \cdot ν &=g_2(x,u_1,u_2) && \text{on } \partialΩ, \end{aligned} \end{equation*} where $ν(x)$ is the outer unit normal of $Ω$ at $x \in \partialΩ$, $Δ_{p_i}$ denotes the $p_i$-Laplacian and $g_i\colon \partialΩ\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ are Carathéodory functions that satisfy general growth and structure conditions for $i=1,2$. In the first part we prove the existence of a positive minimal and a negative maximal solution based on an appropriate construction of sub- and supersolution along with a certain behavior of $g_i$ near zero related to the first eigenvalue of the $p_i$-Laplacian with Steklov boundary condition. The second part is related to the existence of a third nontrivial solution by imposing a variational structure, that is, $(g_1,g_2)=\nabla g$ with a smooth function $(s_1,s_2)\mapsto g(x,s_1,s_2)$. By using the variational characterization of the second eigenvalue of the Steklov eigenvalue problem for the $p_i$-Laplacian together with the properties of the related truncated energy functionals, which are in general nonsmooth, we show the existence of a nontrivial solution whose components lie between the components of the positive minimal and the negative maximal solution. |
| title | Elliptic $p$-Laplacian systems with nonlinear boundary condition |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2401.05846 |