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

Abstract

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 $x_{u},$ \[ \lim_{p\rightarrow\infty}\frac{q(p)}{p}=Q\in\left\{ \begin{array} [c]{lll} (0,1) & \mathrm{if} & 0<\beta<\alpha<1\\ (1,\infty) & \mathrm{if} & 0<\alpha<\beta<1 \end{array} \right. \] and \[ \lim_{p\rightarrow\infty}\sqrt[p]{\mu_{p}}>R^{-\alpha}, \] with $R$ denoting the inradius of $\Omega.$