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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University of Szeged
2024-08-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
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| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=11094 |
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| author | Wei-Long Yang Jia-Feng Liao |
| author_facet | Wei-Long Yang Jia-Feng Liao |
| author_sort | Wei-Long Yang |
| collection | DOAJ |
| description | 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 $4<p<6$. |
| format | Article |
| id | doaj-art-52644e36190b42c6822d32e6a531b016 |
| institution | Kabale University |
| issn | 1417-3875 |
| language | English |
| publishDate | 2024-08-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj-art-52644e36190b42c6822d32e6a531b0162025-01-15T21:24:58ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752024-08-0120244211910.14232/ejqtde.2024.1.4211094Ground state sign-changing solution for a logarithmic Kirchhoff-type equation in $\mathbb{R}^{3}$Wei-Long Yang0Jia-Feng LiaoChina West Normal University, Nanchong, P. R. ChinaWe 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 $4<p<6$.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=11094kirchhoff-type equationlogarithmic nonlinearityground state sign-changing solutionvariational methodstopological degree theory |
| spellingShingle | Wei-Long Yang Jia-Feng Liao Ground state sign-changing solution for a logarithmic Kirchhoff-type equation in $\mathbb{R}^{3}$ Electronic Journal of Qualitative Theory of Differential Equations kirchhoff-type equation logarithmic nonlinearity ground state sign-changing solution variational methods topological degree theory |
| title | Ground state sign-changing solution for a logarithmic Kirchhoff-type equation in $\mathbb{R}^{3}$ |
| title_full | Ground state sign-changing solution for a logarithmic Kirchhoff-type equation in $\mathbb{R}^{3}$ |
| title_fullStr | Ground state sign-changing solution for a logarithmic Kirchhoff-type equation in $\mathbb{R}^{3}$ |
| title_full_unstemmed | Ground state sign-changing solution for a logarithmic Kirchhoff-type equation in $\mathbb{R}^{3}$ |
| title_short | Ground state sign-changing solution for a logarithmic Kirchhoff-type equation in $\mathbb{R}^{3}$ |
| title_sort | ground state sign changing solution for a logarithmic kirchhoff type equation in mathbb r 3 |
| topic | kirchhoff-type equation logarithmic nonlinearity ground state sign-changing solution variational methods topological degree theory |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=11094 |
| work_keys_str_mv | AT weilongyang groundstatesignchangingsolutionforalogarithmickirchhofftypeequationinmathbbr3 AT jiafengliao groundstatesignchangingsolutionforalogarithmickirchhofftypeequationinmathbbr3 |