Sign-changing solution for an elliptic equation with critical growth at the boundary

Marcelo F. Furtado, João Pablo Pinheiro da Silva, Karla Carolina V. De Sousa
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引用次数: 0

Abstract

We prove the existence of sign-changing solution to the problem

$$\begin{aligned} -\Delta u-\dfrac{1}{2}\left( x\cdot \nabla u\right) =\lambda u, \hbox { in }\mathbb {R}_{+}^{N}, \qquad \dfrac{\partial u}{\partial \nu }=|u|^{2_*-2}u, \hbox { on } \partial \mathbb {R}_{+}^{N}, \end{aligned}$$

where \(\mathbb {R}^N_+ = \{(x',x_N): x' \in \mathbb {R}^{N-1},\,x_N>0 \}\) is the upper half-space, \(2_*:=2(N-1)/(N-2)\), \(N \ge 7\), \(\frac{\partial u}{\partial \nu }\) is the partial outward normal derivative and the parameter \(\lambda >0\) interacts with the spectrum of the linearized problem. In the proof, we apply variational methods.

边界有临界增长的椭圆方程的符号变化解
We prove the existence of sign changing solution to the problem $$\begin{aligned} -\Delta u-\dfrac{1}{2}\left( x\cdot \nabla u\right) =\lambda u, \hbox { in }\mathbb {R}_{+}^{N}, \qquad \dfrac{partial u}{\partial \nu }=|u|^{2_*-2}u, \hbox { on }.\partial \mathbb {R}_{+}^{N}, \end{aligned}$$ 其中 \(\mathbb {R}^N_+ = \{(x',x_N): x' \in \mathbb {R}^{N-1},\,x_N>0 \}\)是上半空间, \(2_*:=2(N-1)/(N-2)\),\(N \ge 7\),\(\frac{\partial u}{\partial \nu }\) 是部分向外法导数,参数 \(\lambda >0\) 与线性化问题的频谱相互作用。在证明过程中,我们运用了变分法。
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