We consider the critical problem $-\Delta u-\frac{1}{2}\left(x\cdot \nabla u\right)=0\text{ in }{\mathbb{R}}_{+}^N,\frac{\partial u}{\partial \nu }=\lambda {\left|u\right|}^{p-2}u+{\left|u\right|}^{2_{\ast }-2}u\text{ on }\partial {\mathbb{R}}_{+}^N,$where ${\mathbb{R}}_{+}^N=\left\{(x^{\prime },x_N):x^{\prime }\in {\mathbb{R}}^{N-1},x_N>0\right\}$ is the upper half-space, $N\ge 4$, $\nu$ is the outward normal vector at the boundary and $2\le p<2_{\ast }:=2(N-1)∕(N-2)$. Using a variational approach, we obtain nonnegative nonzero solutions according to the value of the parameter $\lambda >0$.

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上半空间的临界边值问题

我们考虑关键问题 $-\Delta 你-\frac{1}{2}\left(X\cdot \nabla 你\right)=0\text{ 在 }{\mathbb{电阻}}_{+}^N,\frac{\partial 你}{\partial \nu }=\lambda {\left|你\right|}^{磷-2}你+{\left|你\right|}^{2_{＊}-2}你\text{ 上 }\partial {\mathbb{电阻}}_{+}^N,$在哪里 ${\mathbb{电阻}}_{+}^N=\left\{(X^{\prime },X_N)：X^{\prime }\in {\mathbb{电阻}}^{N-1},X_N>0\right\}$ 是上半空间， $N\ge 4$, $\nu$ 是边界处的外向法向量，并且 $2\le 磷<2_{＊}：=2(N-1)∕(N-2)$. 使用变分方法，我们根据参数的值获得非负非零解$\lambda >0$.