On elliptic problems with mixed operators and Dirichlet-Neumann boundary conditions
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913595544043520 |
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| author | Mukherjee, Tuhina Sharma, Lovelesh |
| author_facet | Mukherjee, Tuhina Sharma, Lovelesh |
| contents | In this paper, we study the existence, nonexistence and multiplicity of positive solutions to the problem given by
\begin{equation*} \label{1}
\left\{\begin{split} \mathcal{L}u\: &= λu^{q} + u^{p}, \quad u>0 ~~ \text{in} ~Ω,
u&=0~~\text{in} ~~{D^c},
\mathcal{N}_s(u)&=0 ~~\text{in} ~~{Π_2},
\frac{\partial u}{\partial ν}&=0 ~~\text{in}~~ \partial Ω\cap \overline{Π_2}.
\end{split} \right.\tag{$P_λ$} \end{equation*}
{where $D= \left(Ω\cup {Π_2} \cup (\partialΩ\cap\overline{Π_2})\right)$ and $D^c$ is the complement of $D$, $Ω\subseteq \mathbb{R}^n$ is a non empty open set, $Π_{1}$, $Π_{2}$ are open subsets of $\mathbb{R}^n\setminus{\bar Ω}$ such that $\overline{Π_{1} \cup {Π_{2}}}= \mathbb{R}^n\setminusΩ$, $Π_{1} \cap Π_{2}= \emptyset$ and $Ω\cup Π_2$ is a bounded set with smooth boundary}, $λ>0$ is a real parameter, $ 0 < q < 1<p $, $n>2$ and
$\mathcal{L}= -Δ+(-Δ)^{s},~ \text{for}~s \in (0, 1).$
We first present a functional setting to study any problem involving $\mathcal L$ under mixed boundary conditions in the presence of concave-convex power nonlinearity, {for a suitable range of $λ$, $q$ and $p$}. Our article also contains results related to Picone's identity, strong maximum principles and comparison principles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_02567 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On elliptic problems with mixed operators and Dirichlet-Neumann boundary conditions Mukherjee, Tuhina Sharma, Lovelesh Analysis of PDEs 35A15, 35J25, 35J20 In this paper, we study the existence, nonexistence and multiplicity of positive solutions to the problem given by \begin{equation*} \label{1} \left\{\begin{split} \mathcal{L}u\: &= λu^{q} + u^{p}, \quad u>0 ~~ \text{in} ~Ω, u&=0~~\text{in} ~~{D^c}, \mathcal{N}_s(u)&=0 ~~\text{in} ~~{Π_2}, \frac{\partial u}{\partial ν}&=0 ~~\text{in}~~ \partial Ω\cap \overline{Π_2}. \end{split} \right.\tag{$P_λ$} \end{equation*} {where $D= \left(Ω\cup {Π_2} \cup (\partialΩ\cap\overline{Π_2})\right)$ and $D^c$ is the complement of $D$, $Ω\subseteq \mathbb{R}^n$ is a non empty open set, $Π_{1}$, $Π_{2}$ are open subsets of $\mathbb{R}^n\setminus{\bar Ω}$ such that $\overline{Π_{1} \cup {Π_{2}}}= \mathbb{R}^n\setminusΩ$, $Π_{1} \cap Π_{2}= \emptyset$ and $Ω\cup Π_2$ is a bounded set with smooth boundary}, $λ>0$ is a real parameter, $ 0 < q < 1<p $, $n>2$ and $\mathcal{L}= -Δ+(-Δ)^{s},~ \text{for}~s \in (0, 1).$ We first present a functional setting to study any problem involving $\mathcal L$ under mixed boundary conditions in the presence of concave-convex power nonlinearity, {for a suitable range of $λ$, $q$ and $p$}. Our article also contains results related to Picone's identity, strong maximum principles and comparison principles. |
| title | On elliptic problems with mixed operators and Dirichlet-Neumann boundary conditions |
| topic | Analysis of PDEs 35A15, 35J25, 35J20 |
| url | https://arxiv.org/abs/2311.02567 |