Normalized Solutions to the Kirchhoff-Choquard Equations with Combined Growth
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915055101018112 |
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| author | Goel, Divya Gupta, Shilpa |
| author_facet | Goel, Divya Gupta, Shilpa |
| contents | This paper is devoted to the study of the following nonlocal equation: \begin{equation*} -\left(a+b\|\nabla u\|_{2}^{2(θ-1)}\right) Δu =λu+α(I_μ\ast|u|^{q})|u|^{q-2}u+(I_μ\ast|u|^{p})|u|^{p-2}u \ \hbox{in} \ \mathbb{R}^{N}, \end{equation*} with the prescribed norm $ \int_{\mathbb{R}^{N}} |u|^{2}= c^2,$ where $N\geq 3$, $0<μ<N$, $a,b,c>0$, $1<θ<\frac{2N-μ}{N-2}$, $\frac{2N-μ}{N}<q<p\leq \frac{2N-μ}{N-2}$, $α>0$ is a suitably small real parameter, $λ\in\mathbb{R}$ is the unknown parameter which appears as the Lagrange's multiplier and $I_μ$ is the Riesz potential. We establish existence and multiplicity results and further demonstrate the existence of ground state solutions under the suitable range of $α$. We demonstrate the existence of solution in the case of $q$ is $L^2-$supercritical and $p= \frac{2N-μ}{N-2}$, which is not investigated in the literature till now. In addition, we present certain asymptotic properties of the solutions. To establish the existence results, we rely on variational methods, with a particular focus on the mountain pass theorem, the min-max principle, and Ekeland's variational principle. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_06722 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Normalized Solutions to the Kirchhoff-Choquard Equations with Combined Growth Goel, Divya Gupta, Shilpa Analysis of PDEs 35A15, 35J20, 35J60 This paper is devoted to the study of the following nonlocal equation: \begin{equation*} -\left(a+b\|\nabla u\|_{2}^{2(θ-1)}\right) Δu =λu+α(I_μ\ast|u|^{q})|u|^{q-2}u+(I_μ\ast|u|^{p})|u|^{p-2}u \ \hbox{in} \ \mathbb{R}^{N}, \end{equation*} with the prescribed norm $ \int_{\mathbb{R}^{N}} |u|^{2}= c^2,$ where $N\geq 3$, $0<μ<N$, $a,b,c>0$, $1<θ<\frac{2N-μ}{N-2}$, $\frac{2N-μ}{N}<q<p\leq \frac{2N-μ}{N-2}$, $α>0$ is a suitably small real parameter, $λ\in\mathbb{R}$ is the unknown parameter which appears as the Lagrange's multiplier and $I_μ$ is the Riesz potential. We establish existence and multiplicity results and further demonstrate the existence of ground state solutions under the suitable range of $α$. We demonstrate the existence of solution in the case of $q$ is $L^2-$supercritical and $p= \frac{2N-μ}{N-2}$, which is not investigated in the literature till now. In addition, we present certain asymptotic properties of the solutions. To establish the existence results, we rely on variational methods, with a particular focus on the mountain pass theorem, the min-max principle, and Ekeland's variational principle. |
| title | Normalized Solutions to the Kirchhoff-Choquard Equations with Combined Growth |
| topic | Analysis of PDEs 35A15, 35J20, 35J60 |
| url | https://arxiv.org/abs/2412.06722 |