On the behavior of least energy solutions of a fractional \((p,q(p))\)-Laplacian problem as p goes to infinity

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