On the existence of sign changing bound state solutions of a quasilinear equation

TL;DR

This paper proves the existence of sign-changing bound state solutions of a quasilinear elliptic equation in R^N, with solutions having a prescribed number of zeros, by extending previous techniques under a stronger subcritical condition.

Contribution

It introduces a method to construct sign-changing bound states with a specified number of zeros for a class of quasilinear equations, extending prior ground state existence results.

Findings

Existence of bound state solutions with any given number of zeros.

Extension of techniques to sign-changing solutions under stronger subcritical assumptions.

Application to a class of quasilinear equations involving the m-Laplacian.

Abstract

In this paper we establish the existence of bound state solutions of any given order to $$ \Delta_m u +f(u)=0, x\in R^N, N\ge m>1, (P) $$ where $\Delta_m u=\nabla\cdot(|\nabla u|^{m-2}\nabla u)$ using the same techniques as in [GST] to establish the existence of a ground state solution to (P). Since our solutions change sign, we assume f is continuous in R. The main point here is that by asking a stronger subcritical assumption (see (f_3)(ii) below) than the one considered in [GST], we are able to adapt their techniques to obtain the existence of bound states with a prescribed number of zeros.