| Summary: | 碩士 === 國立清華大學 === 數學系 === 87 === 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 $\Bbb D_s^r$.
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