Abstract
Chen-Lee-Wang [CLW], Chen-Wang [CW], and Lien-Tzeng-Wang [LTW] asserted the existence of a ground state solution of equation (UD) in interior flask domains $\Bbb D_s^r$ : there exists $s_0>0$ such that the index $\alpha (\Bbb D_s^r)$ admits a ground state solution if $s>s_0$, but $\alpha (\Bbb D_s^r)$ does not admit any ground state solution if $s<s_0.$ Is the Esteban-Lions domain $\Bbb D_r^r$ sharp for the existence of solutions: $s_0=r$? In this article, we answer this question partially: there exists a ground state solution of equation (UDf) in a flat interior flask domain : the Esteban-Lions domain $\Bbb S_0^r$ by adding an arbitrary small width but sufficient long corridor. In order to assert our main result, we establish an index comparison criterion: if $\alpha (\Omega) <\alpha (\tilde \Omega_n)$ for some $n$, then there is a ground state solution of equation (UDf) in $\Omega$. We also establish the asymptotic behavior and the symmetry of each solution of equation (UDf) in the interior flask domain $\Bbb D_s^r$.