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), \ \ & \mbox{in}\ \ \Omega, \ \ i = 1, \ldots, k , \\ u_1 = u_2 = \cdots = u_k = 0, \ \ & \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>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>0 $\end{document}</tex-math></inline-formula> is small.</p>

Kirchhoff Biharmonic System with Choquard Nonlinearity and Singular Weights

“…Actually new problems involved with Kirchhoff type emerged from the above researches and authors obtained the existence results of solutions for the Kirchhoff type equations involving critical exponential nonlinearities via variational methods. In relation to Kirchhoff problems, the existence and multiplicity of solutions for elliptic equations inolving critical exponential nonlinearity can be found in the literatures [20,21] and a class of elliptic equations with a small nonhomogeneous term was studied in the articles [22,23,24] in a bounded domain of R 2 . For biharmonic equations we refer to [25,26] in the whole Euclidean space R 4 .…”

“…where Ω is a bounded domain, n ≥ 2, 0 < µ < n, m : R + → R + is a continuous function, ∆nu := div(|∇u| n−2 ∇u), F satisfies suitable growth assumptions and f1 = ∂F ∂u , f1 = ∂F ∂v . Furthermore, in [23], the authors extended to k(∈ N) equations for singular Trudinger-Moser growth in Ω which is a smooth bounded domain in R 2 containing the origin with smooth boundary. And they obtained the multiplicity of solutions of the following elliptic Kirchhoff system…”