Asymptotic behavior as $p\rightarrow\infty$ of least energy solutions of a $(p,q(p))$-Laplacian problem

Abstract

\[ \left\{ \begin{array} [c]{lll} -\left( \Delta_{p}+\Delta_{q(p)}\right) u=\lambda_{p}\left\vert u(x_{u})\right\vert ^{p-2}u(x_{u})\delta_{x_{u}} & \mathrm{in} & \Omega\\ u=0 & \mathrm{on} & \partial\Omega, \end{array} \right. \] where $x_{u}$ is the (unique) maximum point of $\left\vert u\right\vert ,$ $\delta_{x_{u}}$ is the Dirac delta distribution supported at $x_{u},$ \[ \lim_{p\rightarrow\infty}\frac{q(p)}{p}=Q\in\left\{ \begin{array} [c]{lll} (0,1) & \mathrm{if} & N<q(p)<p\\ (1,\infty) & \mathrm{if} & N<p<q(p) \end{array} \right. \] and $\lambda_{p}>0$ is such that \[ \min\left\{ \frac{\left\Vert \nabla u\right\Vert _{\infty}}{\left\Vert u\right\Vert _{\infty}}:0\not \equiv u\in W^{1,\infty}(\Omega)\cap C_{0}(\overline{\Omega})\right\} \leq\lim_{p\rightarrow\infty}(\lambda _{p})^{\frac{1}{p}}<\infty. \]