Existence of sign changing solutions for an equation with a weighted p-Laplace operator

TL;DR

This paper proves the existence of sign-changing radial solutions with any prescribed number of nodes for a weighted p-Laplace elliptic equation using a shooting method and phase plane analysis.

Contribution

It introduces a novel shooting method combined with phase plane change of variables to establish solutions with multiple nodes for weighted p-Laplace equations.

Findings

Solutions with any number of nodes exist for the equation.

The size of the support increases with the number of nodes.

The method applies to solutions with or without compact support.

Abstract

We consider radial solutions of a general elliptic equation involving a weighted $p$-Laplace operator with a subcritical nonlinearity. By a shooting method we prove the existence of solutions with any prescribed number of nodes. The method is based on a change of variables in the phase plane, a very general computation of an angular velocity and new estimates for the decay of an energy associated with an asymptotic Hamiltonian problem. Estimating the rate of decay for the energy requires a sub-criticality condition. The method covers the case of solutions which are not compactly supported or which have compact support. In the last case, we show that the size of the support increases with the number of nodes.