On finite Morse index solutions of higher order fractional elliptic equations

Abstract

<p style='text-indent:20px;'>We consider sign-changing solutions of the equation <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ (-{\Delta})^s u+\lambda u = |u|^{p-1}u \; \mbox{in}\; \mathbb R^n, $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ n\geq 1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ \lambda&gt;0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ p&gt;1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ 1&lt;s\leq2 $\end{document}</tex-math></inline-formula>. The main goal of this work is to analyze the influence of the linear term <inline-formula><tex-math id="M5">\begin{document}$ \lambda u $\end{document}</tex-math></inline-formula>, in order to classify stable solutions possibly unbounded and sign-changing. We prove Liouville type theorems for stable solutions or solutions which are stable outside a compact set of <inline-formula><tex-math id="M6">\begin{document}$ \mathbb R^n $\end{document}</tex-math></inline-formula>. We first derive a monotonicity formula for our equation. After that, we provide integral estimate from stability which combined with Pohozaev-type identity to obtain nonexistence results in the subcritical case with the restrictive condition <inline-formula><tex-math id="M7">\begin{document}$ |u|_{L^{\infty}( \mathbb R^n)}^{p-1}&lt; \frac{\lambda (p+1) }{2} $\end{document}</tex-math></inline-formula>. The supercritical case needs more involved analysis, motivated by the monotonicity formula, we then reduce the nonexistence of nontrivial entire solutions which are stable outside a compact set of <inline-formula><tex-math id="M8">\begin{document}$ \mathbb R^n $\end{document}</tex-math></inline-formula>. Through this approach we give a complete classification of stable solutions <b>for all <inline-formula><tex-math id="M9">\begin{document}$ p&gt;1 $\end{document}</tex-math></inline-formula></b>. Moreover, for the case <inline-formula><tex-math id="M10">\begin{document}$ 0&lt;s\leq1 $\end{document}</tex-math></inline-formula>, finite Morse index solutions are classified in [<xref ref-type="bibr" rid="b19">19</xref>,<xref ref-type="bibr" rid="b25">25</xref>].