A positive solution for an anisotropic $ (p,q) $-Laplacian

Abstract

<p style='text-indent:20px;'>Here, the anisotropic <inline-formula><tex-math id="M2">\begin{document}$ (p, q) $\end{document}</tex-math></inline-formula>-Laplacian</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ - \sum\limits_{i = 1}^N\frac{\partial}{\partial x_i}\left( \left|\frac{\partial u}{\partial x_i}\right|^{p_i-2}\frac{\partial u}{\partial x_i}\right) - \sum\limits_{i = 1}^N\frac{\partial}{\partial x_i}\left( \left|\frac{\partial u}{\partial x_i}\right|^{q_i-2}\frac{\partial u}{\partial x_i}\right) = \lambda u^{\gamma-1} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>is considered, where <inline-formula><tex-math id="M3">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a bounded and regular domain of <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ q_i\leq p_i $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M6">\begin{document}$ i = 1, \cdots, N $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ \gamma &gt; 1 $\end{document}</tex-math></inline-formula>. The existence of positive solution is proved via sub-supersolution method.</p>