On a nonhomogeneous Kirchhoff type elliptic system with the singular Trudinger-Moser growth

Abstract

<p style='text-indent:20px;'>The aim of this paper is to study the multiplicity of solutions for the following Kirchhoff type elliptic systems</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{ll} -m\left(\mathop \sum \limits_{j = 1}^k \|u_j\|^2\right)\Delta u_i = \frac{f_i(x, u_1, \ldots, u_k)}{|x|^\beta}+\varepsilon h_i(x), \ \ &amp; \mbox{in}\ \ \Omega, \ \ i = 1, \ldots, k , \\ u_1 = u_2 = \cdots = u_k = 0, \ \ &amp; \mbox{on}\ \ \partial\Omega, \end{array} \right. \end{eqnarray*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a bounded domain in <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^2 $\end{document}</tex-math></inline-formula> containing the origin with smooth boundary, <inline-formula><tex-math id="M3">\begin{document}$ \beta\in [0, 2) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ m $\end{document}</tex-math></inline-formula> is a Kirchhoff type function, <inline-formula><tex-math id="M5">\begin{document}$ \|u_j\|^2 = \int_\Omega|\nabla u_j|^2dx $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ f_i $\end{document}</tex-math></inline-formula> behaves like <inline-formula><tex-math id="M7">\begin{document}$ e^{\alpha_0 s^2} $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M8">\begin{document}$ |s|\rightarrow \infty $\end{document}</tex-math></inline-formula> for some <inline-formula><tex-math id="M9">\begin{document}$ \alpha_0&gt;0 $\end{document}</tex-math></inline-formula>, and there is <inline-formula><tex-math id="M10">\begin{document}$ C^1 $\end{document}</tex-math></inline-formula> function <inline-formula><tex-math id="M11">\begin{document}$ F: \Omega\times\mathbb{R}^k\to \mathbb{R} $\end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id="M12">\begin{document}$ \left(\frac{\partial F}{\partial u_1}, \ldots, \frac{\partial F}{\partial u_k}\right) = \left(f_1, \ldots, f_k\right) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M13">\begin{document}$ h_i\in \left(\big(H^1_0(\Omega)\big)^*, \|\cdot\|_*\right) $\end{document}</tex-math></inline-formula>. We establish sufficient conditions for the multiplicity of solutions of the above system by using variational methods with a suitable singular Trudinger-Moser inequality when <inline-formula><tex-math id="M14">\begin{document}$ \varepsilon&gt;0 $\end{document}</tex-math></inline-formula> is small.</p>