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

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.