We consider the following problem involving supercritical exponent and polyharmonic operator: $$\begin{aligned} (-\Delta )^mu=K(|y|)u^{m^*-1+\varepsilon }, \;u>0, \hbox { in } B_1(0), \; u \in {\mathcal {D}}_0^{m,2}(B_1(0)), \end{aligned}$$ where $$B_1(0)$$ is the unit ball in $${\mathbb {R}}^{N}$$ , $$m^*=\frac{2N}{N-2m}$$ is the critical exponent, $$\; N\ge 2m+2$$ , $$ m \in {\mathbb {N}}_+$$ , $$\varepsilon > 0$$ , K(|y|) is a nonnegative bounded function. We prove that if $$\varepsilon > 0$$ is small enough, this problem has large number of bubble solutions, and the number of its bubbles varies with the parameter $$\varepsilon $$ at the order $$\varepsilon ^{-1/(N-2m+1)}$$ as $$\varepsilon \rightarrow 0^+$$ . Moreover, all bubbles of the solutions approach the boundary of $$B_1(0)$$ as $$\varepsilon $$ goes to $$0^+$$ .

中文翻译：

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超临界多谐Hénon型方程的气泡解

我们考虑以下涉及超临界指数和多谐算子的问题： $$\begin{aligned} (-\Delta )^mu=K(|y|)u^{m^*-1+\varepsilon }, \;u> 0, \hbox { in } B_1(0), \; u \in {\mathcal {D}}_0^{m,2}(B_1(0)), \end{aligned}$$ 其中 $$B_1(0)$$ 是 $${\mathbb 中的单位球{R}}^{N}$$ , $$m^*=\frac{2N}{N-2m}$$ 是临界指数，$$\；N\ge 2m+2$$ , $$ m \in {\mathbb {N}}_+$$ , $$\varepsilon > 0$$ , K(|y|) 是一个非负有界函数。我们证明如果$$\varepsilon > 0$$足够小，这个问题有大量的气泡解，并且它的气泡数量随着参数$$\varepsilon $$以$$\varepsilon ^{的顺序变化-1/(N-2m+1)}$$ 作为 $$\varepsilon \rightarrow 0^+$$ 。此外，随着 $$\varepsilon $$ 接近 $$0^+$$ ，解的所有气泡都接近 $$B_1(0)$$ 的边界。