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

## TL;DR

This paper investigates the existence and Morse index of least energy nodal solutions for a quasilinear $(p,2)$-Laplacian problem, providing new insights especially when the parameter epsilon equals zero.

## Contribution

It offers a comprehensive review of the existence of least energy nodal solutions and introduces new Morse index results, notably for the case when epsilon is zero.

## Key findings

- Existence of least energy nodal solutions for the quasilinear problem.
- Morse index characterization of these solutions.
- Specific Morse index results for the case epsilon=0.

## Abstract

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 $\ep =0$.