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|...

Full description

Bibliographic Details
Main Author: Xiaolong Yang
Format: Article
Language:English
Published: University of Szeged 2021-04-01
Series:Electronic Journal of Qualitative Theory of Differential Equations
Subjects:
Online Access:http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=8923
Description
Summary: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.
ISSN:1417-3875