Existence and Morse Index of least energy nodal solution of the $(p,2)$-laplacian

In this paper we study the quasilinear equation $- ep^2 Delta u-Delta_p u=f(u)$ in a smooth bounded domain $Omega$ with Dirichlet boundary condition. For $ep geq 0$, we review existence of a least energy nodal solution and then present information about the Morse Index of least nodal energy solutions this BVP. In particular we provide Morse Index information for the case $ep =0$.