We prove a weighted Sobolev trace embedding in the upper half‐space and give its best constant. This embedding can be employed to study a number of critical boundary problems. In this direction, we obtain existence and nonexistence results for a class of semilinear elliptic equations with nonlinear boundary conditions involving critical growth. These equations are closely related to the study of self‐similar solutions for nonlinear reaction‐diffusion equations.

中文翻译：

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关于加权轨迹嵌入和对关键边界问题的应用†

我们证明了加权的Sobolev迹线嵌入上半空间并给出其最佳常数。该嵌入可用于研究许多关键边界问题。在这个方向上，我们获得了一类具有临界增长的非线性边界条件的半线性椭圆型方程的存在性和不存在性。这些方程与非线性反应扩散方程自相似解的研究密切相关。