On the Morse index of least energy nodal solutions for quasilinear elliptic problems

Abstract

In this paper we study the quasilinear equation $$- \varepsilon ^2 \varDelta u-\varDelta _p u=f(u)$$ in a smooth bounded domain $$\varOmega \subset {\mathbb {R}}^N$$ with Dirichlet boundary condition, where $$p>2$$ and f is a suitable subcritical and p-superlinear function at $$\infty $$. First, for $$\epsilon \ne 0$$ we prove that Morse index is two for every least energy nodal solution. This result is inspired and motivated by previous results by A. Castro, J. Cossio and J. M. Neuberger, and T. Bartsch and T. Weth; and it is connected with a result by S. Cingolani and G. Vannella. Then, for the limit case $$\varepsilon = 0$$ we prove (a) the existence of a least energy nodal solution whose Morse index is two, and (b) Morse index is two for every nodal solution which strictly and locally minimizes the energy functional on the set of sign-changing admissible functions.