Existence of nodal solutions of nonlinear elliptic equations

Abstract

We establish that, for $n\geq3$, the elliptic equation

$$ -\Delta u=\lambda|x|^\mu|u|^{q-2}u+|x|^\nu|u|^{p-2}u $$

on a ball with zero Dirichlet data possesses a pair of nodal radial solutions for all $\lambda>0$ provided that

$$ \mu,\nu>-2,\quad\max\bigg\{2,\frac{n+2\mu+2}{n-2}\bigg\}<q<\frac{2(n+\mu)}{n-2}\quad\text{and}\quad p=\frac{2(n+\nu)}{n-2}. $$

When $q=2$ and $n>2\mu+6$, the same result holds for $\lambda>0$ small. Canonical transformations convert the equation into a quasi-linear elliptic equation and an equation with Hardy term. Then the results correspond to the results for the transformed equations. For example, the equation

$$ -\Delta w-\frac{\chi}{|y|^2}w=\tilde{\lambda}|y|^a|w|^{q-2}w +|y|^\nu|w|^{p-2}w, $$

on a ball with zero Dirichlet data, possesses a pair of nodal radial solutions for all $\tilde\lambda>0$ provided that $a,\nu>-2$ and

$$ \max\bigg\{2,\frac{n+a-\sqrt{\bar{\chi}-\chi}}{\sqrt{\bar{\chi}}}\bigg\} <q<\frac{n+a}{\sqrt{\bar{\chi}}}\quad\text{with~}\bar{\chi}=\bigg(\frac{n-2}{2}\bigg)^2. $$

When $q=2$, $n>2a+6$ and $0<\chi<\bar\chi-(a+2)^2$, the same result holds for $\tilde\lambda>0$ small.