In this paper, we study a class of semilinear $p\left(x\right)$-Laplacian equations with variable exponent sources. By using potential well method, we first prove a threshold results on the existence and nonexistence of global solutions to the equations when initial energy is less than the mountain pass level $d$. In the former case we also show the decay properties of energy functional. We finally obtain the non-global existence results with high energy initial data.

中文翻译：

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势阱法 $p（X）$-具有可变指数源的拉普拉斯方程

在本文中，我们研究了一类半线性 $p（X）$-具有可变指数源的拉普拉斯方程。通过使用势阱方法，我们首先证明了当初始能量小于山口通过水平时，方程整体解的存在与否的阈值结果$d$。在前一种情况下，我们还显示了能量函数的衰减特性。我们最终获得了具有高能量初始数据的非全局存在结果。