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.

中文翻译：

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拟线性椭圆问题的最小能量节点解的摩尔斯指数

摘要

在本文中，我们研究了在光滑有界域\（\ varOmega \ subset {\ mathbb {R}} ^ N中的拟线性方程\（-\ varepsilon ^ 2 \ varDelta u- \ varDelta _p u = f（u）\）\）具有Dirichlet边界条件，其中\（p> 2 \）和f是\（\ infty \）处合适的亚临界和p-超线性函数。首先，对于\（\ epsilon \ ne 0 \），我们证明对于每个最小能级节点解，莫尔斯指数为2。这个结果是由A. Castro，J。Cossio和JM Neuberger以及T. Bartsch和T. Weth的先前结果启发和激发的。它与S. Cingolani和G. Vannella的结果相关。然后，对于极限情况\（\ varepsilon = 0 \） 我们证明（a）存在一个最低能量的节点解，其摩尔斯指数为2，并且（b）每个节点解的摩尔斯指数为2，这严格且局部地使符号改变的可允许函数集上的能量函数最小。