Infinitely many nodal solutions for a class of quasilinear elliptic equations
In this paper, we study the existence of infinitely many nodal solutions for the following quasilinear elliptic equation \begin{equation*} \begin{cases} -\nabla\cdot\left[\phi'(|\nabla u|^2)\nabla u\right]+|u|^{\alpha-2}u=f(u) , \quad x\in \mathbb{R}^N,\\ u(x)\rightarrow 0, \quad \mbox{as} ~|x|...
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| Format: | Article |
| Language: | English |
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University of Szeged
2021-04-01
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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=8923 |
| _version_ | 1827951084787728384 |
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| author | Xiaolong Yang |
| author_facet | Xiaolong Yang |
| author_sort | Xiaolong Yang |
| collection | DOAJ |
| description | In this paper, we study the existence of infinitely many
nodal solutions for the following quasilinear elliptic equation
\begin{equation*}
\begin{cases}
-\nabla\cdot\left[\phi'(|\nabla u|^2)\nabla u\right]+|u|^{\alpha-2}u=f(u) , \quad x\in \mathbb{R}^N,\\
u(x)\rightarrow 0, \quad \mbox{as} ~|x|\rightarrow \infty,
\end{cases}
\end{equation*}
where $N\geq2$, $\phi(t)$ behaves like $t^{q/2}$ for small $t$ and $t^{p/2}$ for large $t$, $1<p<q<N$, $f\in \mathcal{C}^1(\mathbb{R}^+,\mathbb{R})$ is of subcritical, $q\le\alpha\le p^{*}q'/p'$, let $p^{*}=\frac{Np}{N-p}$, $p'$ and $q'$ be the conjugate exponents respectively of $p$ and $q$. For any given integer $k\geq 0$, we prove that the equation has a pair of radial nodal solution with exactly $k$ nodes. |
| first_indexed | 2024-04-09T13:37:16Z |
| format | Article |
| id | doaj.art-6006bf7511d8458383ff7936dc2bfc18 |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:37:16Z |
| publishDate | 2021-04-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-6006bf7511d8458383ff7936dc2bfc182023-05-09T07:53:11ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752021-04-0120213212010.14232/ejqtde.2021.1.328923Infinitely many nodal solutions for a class of quasilinear elliptic equationsXiaolong Yang0Central China Normal University, Wuhan, P. R. ChinaIn this paper, we study the existence of infinitely many nodal solutions for the following quasilinear elliptic equation \begin{equation*} \begin{cases} -\nabla\cdot\left[\phi'(|\nabla u|^2)\nabla u\right]+|u|^{\alpha-2}u=f(u) , \quad x\in \mathbb{R}^N,\\ u(x)\rightarrow 0, \quad \mbox{as} ~|x|\rightarrow \infty, \end{cases} \end{equation*} where $N\geq2$, $\phi(t)$ behaves like $t^{q/2}$ for small $t$ and $t^{p/2}$ for large $t$, $1<p<q<N$, $f\in \mathcal{C}^1(\mathbb{R}^+,\mathbb{R})$ is of subcritical, $q\le\alpha\le p^{*}q'/p'$, let $p^{*}=\frac{Np}{N-p}$, $p'$ and $q'$ be the conjugate exponents respectively of $p$ and $q$. For any given integer $k\geq 0$, we prove that the equation has a pair of radial nodal solution with exactly $k$ nodes.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=8923quasilinear elliptic equationnodal solutionsmultiple solutions |
| spellingShingle | Xiaolong Yang Infinitely many nodal solutions for a class of quasilinear elliptic equations Electronic Journal of Qualitative Theory of Differential Equations quasilinear elliptic equation nodal solutions multiple solutions |
| title | Infinitely many nodal solutions for a class of quasilinear elliptic equations |
| title_full | Infinitely many nodal solutions for a class of quasilinear elliptic equations |
| title_fullStr | Infinitely many nodal solutions for a class of quasilinear elliptic equations |
| title_full_unstemmed | Infinitely many nodal solutions for a class of quasilinear elliptic equations |
| title_short | Infinitely many nodal solutions for a class of quasilinear elliptic equations |
| title_sort | infinitely many nodal solutions for a class of quasilinear elliptic equations |
| topic | quasilinear elliptic equation nodal solutions multiple solutions |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=8923 |
| work_keys_str_mv | AT xiaolongyang infinitelymanynodalsolutionsforaclassofquasilinearellipticequations |