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 \[ \left\{\begin{array} [c]{lll} \left[ \left( -Δ_{p}\right) ^α+\left( -Δ_{q(p)}\right) ^β\right] u=μ_{p}\left\Vert u\right\Vert _{\infty}^{p-2} u(x_{u})δ_{x_{u}} & \mathrm{in} & Ω\\ u=0 & \mathrm{in} & \mathbb{R}^{N}\setminusΩ\\ \left\vert u(x_{u})\right\vert =\left\Vert u\right\Vert _{\infty}, & & \end{array} \right. \] where $Ω\subset\mathbb{R}^{N}$ is a bounded, smooth domain, $δ_{x_{u}}$ is the Dirac delta distribution supported at $x_{u},$ \[ \lim_{p\rightarrow\infty}\frac{q(p)}{p}=Q\in\left\{ \begin{array} [c]{lll} (0,1) & \mathrm{if} & 0<β<α<1\\ (1,\infty) & \mathrm{if} & 0<α<β<1 \end{array} \right. \] and \[ \lim_{p\rightarrow\infty}\sqrt[p]{μ_{p}}>R^{-α}, \] with $R$ denoting the inradius of $Ω.$