Positive solutions to a nonlinear fractional equation with an external source term

Abstract

<p style='text-indent:20px;'>This paper deals with the following nonlinear fractional equation with an external source term</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \label{eqS0.1} (-\Delta)^{s}u +u = K(x)u^{p}+f(x), \; u&gt;0, \; x\in{\Bbb R}^N, \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ N&gt;2s $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ 0&lt;s&lt;1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ 1&lt;p&lt;2_{\ast}(s)-1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ 2_{\ast}(s) = \frac{2N}{N-2s} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ K(x) $\end{document}</tex-math></inline-formula> is a continuous function and <inline-formula><tex-math id="M6">\begin{document}$ f\in L^{2}({\Bbb R}^{N})\cap L^{\infty}({\Bbb R}^{N}) $\end{document}</tex-math></inline-formula>. Using a Lyapunov-Schmidt reduction scheme, we prove that the equation admits <inline-formula><tex-math id="M7">\begin{document}$ k $\end{document}</tex-math></inline-formula>-peak solutions for any integer <inline-formula><tex-math id="M8">\begin{document}$ k&gt;0 $\end{document}</tex-math></inline-formula> if <inline-formula><tex-math id="M9">\begin{document}$ f $\end{document}</tex-math></inline-formula> is small and <inline-formula><tex-math id="M10">\begin{document}$ K(x) $\end{document}</tex-math></inline-formula> satisfies some additional assumptions at infinity. The main difficulty is to improve the estimate of the remainder obtained in the reduction process.</p>