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LetΩbe a smooth bounded domain in RN, we consider the problem withλ<0,1<r<1*. We consider the problem (P) as the limit case of a sequence of pn Laplace equations when pn tends to 1, and study the mountain pass type solution of pn Laplace equations, applying concentration compactness theorem and Radon-Nikodym theorem we get a nontrivial solution of problem (P) in W01,1(Ω). This thesis consists of four chapters.In Chapter one, we introduce the background and progress of the 1-Laplace equation, and give the research questions and main results of this paper;In Chapter two, we give some preliminary results and definitions that will be used in this thesis;In Chapter three, we mainly study the existence of the nontrivial solution of the 1-Laplace equation in W01,1(Ω);In Chapter four, we summarize the main results of this thesis and the future direction of the main research.
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