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\[ \left\{ \begin{array} [c]{lll} -\left( \Delta_{p}+\Delta_{q(p)}\right)u=\lambda_{p}\left\vert u(x_{u})\right\vert ^{p-2}u(x_{u})\delta_{x_{u}} &\mathrm{in} & \Omega\\ u=0 & \mathrm{on} & \partial\Omega, \end{array} \right.\] 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 \[ \min\left\{\frac{\left\Vert \nabla u\right\Vert _{\infty}}{\left\Vert u\right\Vert_{\infty}}:0\not \equiv u\in W^{1,\infty}(\Omega)\capC_{0}(\overline{\Omega})\right\} \leq\lim_{p\rightarrow\infty}(\lambda_{p})^{\frac{1}{p}}<\infty. \]

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