Existence of nodal solutions of nonlinear elliptic equations

Abstract

We establish that, for -\Delta u=\lambda|x|^\mu|u|^{q-2}u+|x|^\nu|u|^{p-2}u \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}. -\Delta w-\frac{\chi}{|y|^2}w=\tilde{\lambda}|y|^a|w|^{q-2}w +|y|^\nu|w|^{p-2}w, \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. small.