Fractional $ p $-sub-Laplacian operator problem with concave-convex nonlinearities on homogeneous groups

Abstract

<p style='text-indent:20px;'>This study examines the existence and multiplicity of non-negative solutions of the following fractional <inline-formula><tex-math id="M2">\begin{document}$ p $\end{document}</tex-math></inline-formula>-sub-Laplacian problem</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\{\begin{aligned} &amp;(-\Delta_{p,g})^{s}u = \lambda f(x)|u|^{\alpha-2}u+ h(x)|u|^{\beta-2} u \quad&amp;\rm{in}\,\,\, &amp;\Omega,\\ &amp;\,\,\, u = 0\quad\quad &amp;\rm{in} \,\,\, &amp;\mathbb{G}\setminus \Omega, \end{aligned}\right. \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M3">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is an open bounded in homogeneous Lie group <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{G} $\end{document}</tex-math></inline-formula> with smooth boundary, <inline-formula><tex-math id="M5">\begin{document}$ p&gt;1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ s\in(0,1) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ (-\Delta_{p,g})^{s} $\end{document}</tex-math></inline-formula> is the fractional <inline-formula><tex-math id="M8">\begin{document}$ p $\end{document}</tex-math></inline-formula>-sub-Laplacian operator with respect to the quasi-norm <inline-formula><tex-math id="M9">\begin{document}$ g $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$ \lambda&gt;0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M11">\begin{document}$ 1&lt; \alpha&lt;p &lt;\beta &lt; p^*_{s} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M12">\begin{document}$ p^*_{s}: = \frac{Qp}{Q-sp} $\end{document}</tex-math></inline-formula> is the fractional critical Sobolev exponents, <inline-formula><tex-math id="M13">\begin{document}$ Q $\end{document}</tex-math></inline-formula> is the homogeneous dimensions of the homogeneous Lie group <inline-formula><tex-math id="M14">\begin{document}$ \mathbb{G} $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M15">\begin{document}$ Q&gt; sp $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M16">\begin{document}$ f $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M17">\begin{document}$ h $\end{document}</tex-math></inline-formula> are sign-changing smooth functions. With the help of the Nehari manifold, we prove that the nonlocal problem on homogeneous group has at least two nontrivial solutions when the parameter <inline-formula><tex-math id="M18">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> belong to a center subset of <inline-formula><tex-math id="M19">\begin{document}$ (0,+\infty) $\end{document}</tex-math></inline-formula>.</p>