Ground state sign-changing solution for a logarithmic Kirchhoff-type equation in $\mathbb{R}^{3}$
We investigate the following logarithmic Kirchhoff-type equation: \begin{equation*} \left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}+V(x)u^{2}dx\right)[-\Delta u+V(x)u]=|u|^{p-2}u\ln |u|,\qquad x\in\mathbb{R}^{3}, \end{equation*} where $a,b>0$ are constants, $4<p<2^{*}=6$. Under some appropria...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2024-08-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Subjects: | |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=11094 |
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| Summary: | We investigate the following logarithmic Kirchhoff-type equation:
\begin{equation*}
\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}+V(x)u^{2}dx\right)[-\Delta u+V(x)u]=|u|^{p-2}u\ln |u|,\qquad x\in\mathbb{R}^{3},
\end{equation*}
where $a,b>0$ are constants, $4<p<2^{*}=6$. Under some appropriate hypotheses on the potential function $V$, we prove the existence of a positive ground state solution, a ground state sign-changing solution and a sequence of solutions by using the constraint variational methods, topological degree theory, quantitative deformation lemma and symmetric mountain pass theorem. Our results complete those of Gao et al. [Appl. Math. Lett. 139(2023), 108539] with the case of $4<p<6$. |
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| ISSN: | 1417-3875 |