Nonexistence results on the space or the half space of $ -\Delta u+\lambda u = |u|^{p-1}u $ via the Morse index

In this paper we consider the following semi-linear elliptic problem

$ \begin{equation*} -\Delta u+\lambda u = |u|^{p-1}u\quad\mbox{in}\,\, \mathcal{O}, \tag{P} \end{equation*} $

where $ \mathcal{O} = \mathbb{R}^N $; or $ \mathcal{O} = \mathbb{R}^N_+ = \{x = (x',x_N),\, x'\in \mathbb{R}^{N-1},x_N>0\} $ with Dirichlet boundary conditions. Here $ N\geq2 $, $ p>1 $ and $ \lambda $ is a positive real parameter. The main goal ofthis work is to analyze the influence of the linear term $ \lambda u $, in order to classify regular stable solutions possibly unbounded and sign-changing. Our analysis reveals the nonexistence of nontrivial stable solutions (respectively solutions which are stable outside a compact set) for all $ p> 1 $ (respectively for all $ p\geq \frac{N+2}{N-2} $, or $ 1<p<\frac{N+2}{N-2} $ and $ |u|^{p-1}<\frac{\lambda (p+1)}{2} $). Inspired by [6,9,16,23], we establish a monotonicity formula to discuss the supercritical case.

Regarding the case $ \mathcal{O} = \mathbb{R}^N $, we obtain a complete classification which states that problem $ (P) $ has regular solutions which are stable outside a compact set if and only if $ p\in (1,\infty) $ and $ N = 2 $; or $ p\in(1,\frac{N+2}{N-2}) $ and $ N\geq3. $