在這篇論文中，我們考慮一個半線性橢圓方程式在一些無界定義域非零解的存在性問題。為了找出半線性橢圓方程式的非零解，許多人用了各種不同的方法，而我們則是利用維塔利收斂定理和巴萊斯麥爾理論，也就是有巴萊斯麥爾數列、巴萊斯麥爾條件，藉由函數收斂的緊緻性質來找尋此方程式的非零解。所以在這篇論文中，我們呈現了一些巴萊斯麥爾數列的新結果，而且我們還證明了巴萊斯麥爾條件、維塔利收斂定理、定義域的指數、半線性橢圓方程在一些其它無界定義域正解的存在性問題和之前的人所做的方法是等價的，這樣將有助於我們尋找此半線性橢圓方程式非零解的存在性。

In this article, we present several new results for Palais—Smale sequences. Consequently, we unify the Vitali convergence theorem and many main concepts in the variational methods by Lions, Lien-Tzeng-Wang, del Pino-Felmer and Chabrowski.

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