We study the limiting behavior of the minimizers for the following Kirchhoff energy functional with an ellipse-shaped type trapping potential \(V\left( x \right) \) in a bounded domain \(\Omega \) of \(\mathbb {R} ^{2} \) , where the energy functional is defined by \(\begin{aligned} E_{b} \left( u\right) = {\mathop {\int }\limits _{\Omega }}\left| \nabla u\right| ^{2} \textrm{d}x+{\mathop {\int }\limits _{\Omega }} V(x)\left| u\right| ^{2}\textrm{d}x + \frac{b}{2}\left( {\mathop {\int }\limits _{\Omega }}\left| \nabla u\right| ^{2} \textrm{d}x \right) ^{2} - \frac{a }{2} {\mathop {\int }\limits _{\Omega }}\left| u\right| ^{4}\textrm{d}x,\ u\in K. \end{aligned}\) It has been shown that the minimizers always exist for any \(b> 0\) . In the present paper, we consider the limiting behavior of minimizers when the endpoints of the major axis of the ellipse-shaped bottom locate at the interior or the boundary of \(\Omega \) as \(b\searrow 0\) . We first prove that the minimizers must concentrate at an inner point of \(\Omega \) as \(b\searrow 0\) if one of the endpoints of the major axis of the ellipse-shaped bottom locates at the interior of \(\Omega \) . Besides, if all the endpoints of the major axis of the ellipse-shaped bottom are located at the boundary of \(\Omega \) , we obtain that the minimizers must concentrate near the boundary of \(\Omega \) as \(b\searrow 0\) .
