A multiparameter fractional Laplace problem with semipositone nonlinearity

Abstract

<p style='text-indent:20px;'>In this paper we prove the existence of at least one positive solution for nonlocal semipositone problem of the type</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ (P_\lambda^\mu)\left\{ \begin{array}{rcl} (-\Delta)^s u&amp; = &amp; \lambda(u^{q}-1)+\mu u^r \text{ in } \Omega\\ u&amp;&gt;&amp;0 \text{ in } \Omega\\ u&amp;\equiv &amp;0 \text{ on }{\mathbb R^N\setminus\Omega}. \end{array}\right. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>when the positive parameters <inline-formula><tex-math id="M1">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> belong to certain range. Here <inline-formula><tex-math id="M3">\begin{document}$ \Omega\subset \mathbb{R}^N $\end{document}</tex-math></inline-formula> is assumed to be a bounded open set with smooth boundary, <inline-formula><tex-math id="M4">\begin{document}$ s\in (0, 1), N&gt; 2s $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ 0&lt;q&lt;1&lt;r\leq \frac{N+2s}{N- 2s}. $\end{document}</tex-math></inline-formula> First we consider <inline-formula><tex-math id="M6">\begin{document}$ (P_ \lambda^\mu) $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M7">\begin{document}$ \mu = 0 $\end{document}</tex-math></inline-formula> and prove that there exists <inline-formula><tex-math id="M8">\begin{document}$ \lambda_0\in(0, \infty) $\end{document}</tex-math></inline-formula> such that for all <inline-formula><tex-math id="M9">\begin{document}$ \lambda&gt; \lambda_0 $\end{document}</tex-math></inline-formula> the problem <inline-formula><tex-math id="M10">\begin{document}$ (P_ \lambda^0) $\end{document}</tex-math></inline-formula> admits at least one positive solution. In fact we will show the existence of a continuous branch of maximal solutions of <inline-formula><tex-math id="M11">\begin{document}$ (P_\lambda^0) $\end{document}</tex-math></inline-formula> emanating from infinity. Next for each <inline-formula><tex-math id="M12">\begin{document}$ \lambda&gt;\lambda_0 $\end{document}</tex-math></inline-formula> and for all <inline-formula><tex-math id="M13">\begin{document}$ 0&lt;\mu&lt;\mu_{\lambda} $\end{document}</tex-math></inline-formula> we establish the existence of at least one positive solution of <inline-formula><tex-math id="M14">\begin{document}$ (P_\lambda^\mu) $\end{document}</tex-math></inline-formula> using variational method. Also in the sub critical case, i.e., for <inline-formula><tex-math id="M15">\begin{document}$ 1&lt;r&lt;\frac{N+2s}{N-2s} $\end{document}</tex-math></inline-formula>, we show the existence of second positive solution via mountain pass argument.</p>