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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.00654 |
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| _version_ | 1866911087597715456 |
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| author | Kanungo, Suman Mishra, Pawan Kumar |
| author_facet | Kanungo, Suman Mishra, Pawan Kumar |
| contents | In this paper, we study the following class of weighted Choquard equations
\begin{align*}
-Δu =λu + \Bigg(\displaystyle\int\limits_Ω\frac{Q(|y|)F(u(y))}{|x-y|^μ}dy\Bigg) Q(|x|)f(u) ~~\textrm{in}~~ Ω~~ \text{and}~~
u=0~~ \textrm{on}~~ \partial Ω,
\end{align*}
where $Ω\subset \mathbb{R}^2$ is a bounded domain with smooth boundary, $μ\in (0,2)$ and $λ>0$ is a parameter. We assume that $f$ is a real valued continuous function satisfying critical exponential growth in the Trudinger-Moser sense, and $F$ is the primitive of $f$. Let $Q$ be a positive real valued continuous weight, which can be singular at zero. Our main goal is to prove the existence of a nontrivial solution for all parameter values except the resonant case, i.e., when $λ$ coincides with any of the eigenvalues of the operator $(-Δ, H^1_0(Ω))$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_00654 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Nonlocal problem with critical exponential nonlinearity of convolution type: A non-resonant case Kanungo, Suman Mishra, Pawan Kumar Analysis of PDEs Primary 35J15, 35J20, 35J60 In this paper, we study the following class of weighted Choquard equations \begin{align*} -Δu =λu + \Bigg(\displaystyle\int\limits_Ω\frac{Q(|y|)F(u(y))}{|x-y|^μ}dy\Bigg) Q(|x|)f(u) ~~\textrm{in}~~ Ω~~ \text{and}~~ u=0~~ \textrm{on}~~ \partial Ω, \end{align*} where $Ω\subset \mathbb{R}^2$ is a bounded domain with smooth boundary, $μ\in (0,2)$ and $λ>0$ is a parameter. We assume that $f$ is a real valued continuous function satisfying critical exponential growth in the Trudinger-Moser sense, and $F$ is the primitive of $f$. Let $Q$ be a positive real valued continuous weight, which can be singular at zero. Our main goal is to prove the existence of a nontrivial solution for all parameter values except the resonant case, i.e., when $λ$ coincides with any of the eigenvalues of the operator $(-Δ, H^1_0(Ω))$. |
| title | Nonlocal problem with critical exponential nonlinearity of convolution type: A non-resonant case |
| topic | Analysis of PDEs Primary 35J15, 35J20, 35J60 |
| url | https://arxiv.org/abs/2408.00654 |