On singular problems associated with mixed operators under mixed boundary conditions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912186156187648 |
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| author | Mukherjee, Tuhina Sharma, Lovelesh |
| author_facet | Mukherjee, Tuhina Sharma, Lovelesh |
| contents | In this paper, we study the following singular problem associated with mixed operators (the combination of the classical Laplace operator and the fractional Laplace operator) under mixed boundary conditions \begin{equation*} \label{1}
\left\{
\begin{aligned}
\mathcal{L}u &= g(u), \quad u > 0 \quad \text{in} \quad Ω,
u &= 0 \quad \text{in} \quad U^c,
\mathcal{N}_s(u) &= 0 \quad \text{in} \quad \mathcal{N},
\frac{\partial u}{\partial ν} &= 0 \quad \text{in} \quad \partial Ω\cap \overline{\mathcal{N}},
\end{aligned}
\right.
\tag{$P_λ$} \end{equation*}
where $U= (Ω\cup {\mathcal{N}} \cup (\partialΩ\cap\overline{\mathcal{N}}))$, $Ω\subseteq \mathbb{R}^N$ is a non empty open set, $\mathcal{D}$, $\mathcal{N}$ are open subsets of $\mathbb{R}^N\setminus{\bar{Ω}}$ such that ${\mathcal{D}} \cup {\mathcal{N}}= \mathbb{R}^N\setminus{\barΩ}$, $\mathcal{D} \cap {\mathcal{N}}= \emptyset $ and $Ω\cup \mathcal{N}$ is a bounded set with smooth boundary, $λ>0$ is a real parameter and
$\mathcal{L}= -Δ+(-Δ)^{s},~ \text{for}~s \in (0, 1).$
Here $g(u)=u^{-q}$ or $g(u)= λu^{-q}+ u^p$ with $0<q<1<p\leq 2^*-1$. We study $(P_λ)$ to derive the existence of weak solutions along with its $L^\infty$-regularity. Moreover, some Sobolev-type variational inequalities associated with these weak solutions are established. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_07338 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On singular problems associated with mixed operators under mixed boundary conditions Mukherjee, Tuhina Sharma, Lovelesh Analysis of PDEs 35A01, 35A15, 35R11, 35J25, 35J20, 47G20 In this paper, we study the following singular problem associated with mixed operators (the combination of the classical Laplace operator and the fractional Laplace operator) under mixed boundary conditions \begin{equation*} \label{1} \left\{ \begin{aligned} \mathcal{L}u &= g(u), \quad u > 0 \quad \text{in} \quad Ω, u &= 0 \quad \text{in} \quad U^c, \mathcal{N}_s(u) &= 0 \quad \text{in} \quad \mathcal{N}, \frac{\partial u}{\partial ν} &= 0 \quad \text{in} \quad \partial Ω\cap \overline{\mathcal{N}}, \end{aligned} \right. \tag{$P_λ$} \end{equation*} where $U= (Ω\cup {\mathcal{N}} \cup (\partialΩ\cap\overline{\mathcal{N}}))$, $Ω\subseteq \mathbb{R}^N$ is a non empty open set, $\mathcal{D}$, $\mathcal{N}$ are open subsets of $\mathbb{R}^N\setminus{\bar{Ω}}$ such that ${\mathcal{D}} \cup {\mathcal{N}}= \mathbb{R}^N\setminus{\barΩ}$, $\mathcal{D} \cap {\mathcal{N}}= \emptyset $ and $Ω\cup \mathcal{N}$ is a bounded set with smooth boundary, $λ>0$ is a real parameter and $\mathcal{L}= -Δ+(-Δ)^{s},~ \text{for}~s \in (0, 1).$ Here $g(u)=u^{-q}$ or $g(u)= λu^{-q}+ u^p$ with $0<q<1<p\leq 2^*-1$. We study $(P_λ)$ to derive the existence of weak solutions along with its $L^\infty$-regularity. Moreover, some Sobolev-type variational inequalities associated with these weak solutions are established. |
| title | On singular problems associated with mixed operators under mixed boundary conditions |
| topic | Analysis of PDEs 35A01, 35A15, 35R11, 35J25, 35J20, 47G20 |
| url | https://arxiv.org/abs/2501.07338 |