We study the behavior as $p\rightarrow\infty$ of $u_{p},$ a positive least energy solution of the problem \[ \left\{\begin{array} [c]{lll} \left[ \left( -\Delta_{p}\right) ^{\alpha}+\left( -\Delta_{q(p)}\right) ^{\beta}\right] u=\mu_{p}\left\Vert u\right\Vert _{\infty}^{p-2} u(x_{u})\delta_{x_{u}} & \mathrm{in} & \Omega\\ u=0 & \mathrm{in} & \mathbb{R}^{N}\setminus\Omega\\ \left\vert u(x_{u})\right\vert =\left\Vert u\right\Vert _{\infty}, & & \end{array} \right. \] where $\Omega\subset\mathbb{R}^{N}$ is a bounded, smooth domain, $\delta_{x_{u}}$ is the Dirac delta distribution supported at $x_{u},$ \[ \lim_{p\rightarrow\infty}\frac{q(p)}{p}=Q\in\left\{ \begin{array} [c]{lll} (0,1) & \mathrm{if} & 0 R^{-\alpha}, \] with $R$ denoting the inradius of $\Omega.$

中文翻译：

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关于分数 ( p , q ( p ) )-拉普拉斯问题的最小能量解在 p 趋于无穷大时的行为

我们研究 $u_{p} 的 $p\rightarrow\infty$ 行为，$ 问题的最小能量正解 \[ \left\{\begin{array} [c]{lll} \left[ \left ( -\Delta_{p}\right) ^{\alpha}+\left( -\Delta_{q(p)}\right) ^{\beta}\right] u=\mu_{p}\left\Vert u\right\Vert _{\infty}^{p-2} u(x_{u})\delta_{x_{u}} & \mathrm{in} & \Omega\\ u=0 & \mathrm{in } & \mathbb{R}^{N}\setminus\Omega\\ \left\vert u(x_{u})\right\vert =\left\Vert u\right\Vert _{\infty}, & & \end{array} \right。\] 其中 $\Omega\subset\mathbb{R}^{N}$ 是有界平滑域，$\delta_{x_{u}}$ 是 $x_{u},$ \ 支持的 Dirac delta 分布[ \lim_{p\rightarrow\infty}\frac{q(p)}{p}=Q\in\left\{ \begin{array} [c]{lll} (0,1) & \mathrm{if } & 0 R^{-\alpha}, \] 其中 $R$ 表示 $\Omega.$ 的半径