LAUSR

We consider sign-changing solutions of the equation \begin{document}$ (-{\Delta})^s u+\lambda u = |u|^{p-1}u \; \mbox{in}\; \mathbb R^n, $\end{document} where \begin{document}$ n\geq 1 $\end{document} , \begin{document}$ \lambda>0 $\end{document} , \begin{document}$ p>1 $\end{document} and \begin{document}$ 1 . The main goal of this work is to analyze the influence of the linear term \begin{document}$ \lambda u $\end{document} , in order to classify stable solutions possibly unbounded and sign-changing. We prove Liouville type theorems for stable solutions or solutions which are stable outside a compact set of \begin{document}$ \mathbb R^n $\end{document} . We first derive a monotonicity formula for our equation. After that, we provide integral estimate from stability which combined with Pohozaev-type identity to obtain nonexistence results in the subcritical case with the restrictive condition \begin{document}$ |u|_{L^{\infty}( \mathbb R^n)}^{p-1} . The supercritical case needs more involved analysis, motivated by the monotonicity formula, we then reduce the nonexistence of nontrivial entire solutions which are stable outside a compact set of \begin{document}$ \mathbb R^n $\end{document} . Through this approach we give a complete classification of stable solutions for all \begin{document}$ p>1 $\end{document} . Moreover, for the case \begin{document}$ 0 , finite Morse index solutions are classified in [ 19 , 25 ].