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 $\mathit{p}\to \infty$ of ${\mathit{u}}_{\mathit{p}}$ , a positive least energy solution of the problem $$\begin{array}{c}\left\{\begin{array}{cc}[{\mathrm{(}-{\mathrm{\Delta }}_{\mathit{p}}\mathrm{)}}^{\mathit{\alpha }}+{\mathrm{(}-{\mathrm{\Delta }}_{\mathit{q}\mathrm{(}\mathit{p}\mathrm{)}}\mathrm{)}}^{\mathit{\beta }}]\mathit{u}={\mathit{\mu }}_{\mathit{p}}|\mathit{u}\mathrm{(}{\mathit{x}}_{\mathit{u}}\mathrm{)}{|}^{\mathit{p}-2}\mathit{u}\mathrm{(}{\mathit{x}}_{\mathit{u}}\mathrm{)}{\mathit{\delta }}_{{\mathit{x}}_{\mathit{u}}}\hfill & \text{in }\mathrm{\Omega }\hfill \\ \mathit{u}=0\hfill & \text{in }{\mathbb{R}}^{\mathit{N}}\setminus \mathrm{\Omega }\hfill \\ |\mathit{u}\mathrm{(}{\mathit{x}}_{\mathit{u}}\mathrm{)}|=\Vert \mathit{u}{\Vert }_{\infty }\mathrm{,}\hfill \end{array}\right.\end{array}$$ where $\mathrm{\Omega }\subset {\mathbb{R}}^{\mathit{N}}$ is a bounded, smooth domain, ${\mathit{\delta }}_{{\mathit{x}}_{\mathit{u}}}$ is the Dirac delta distribution supported at ${\mathit{x}}_{\mathit{u}}$ , $$\begin{array}{c}\underset{\mathit{p}\to \infty }{lim}{\displaystyle \frac{\mathit{q}\mathrm{(}\mathit{p}\mathrm{)}}{\mathit{p}}}=\mathit{Q}\in \left\{\begin{array}{cc}\mathrm{(}0\mathrm{,}1\mathrm{)}\hfill & \text{if }0\mathrm{<}\mathit{\beta }\mathrm{<}\mathit{\alpha }\mathrm{<}1\hfill \\ \mathrm{(}1\mathrm{,}\infty \mathrm{)}\hfill & \text{if }0\mathrm{<}\mathit{\alpha }\mathrm{<}\mathit{\beta }\mathrm{<}1\hfill \end{array}\right.\end{array}$$ and $$\begin{array}{c}\underset{\mathit{p}\to \infty }{lim}\sqrt[\mathit{p}]{{\mathit{\mu }}_{\mathit{p}}}\mathrm{>}{\mathit{R}}^{-\mathit{\alpha }}\mathrm{,}\end{array}$$ with R denoting the inradius of Ω.