Asymptotic behavior as of least energy solutions of a -Laplacian problem
arXiv:1710.11287 · doi:10.1017/prm.2018.111
Abstract
\[ \left\{ \begin{array} [c]{lll} -\left( Δ_{p}+Δ_{q(p)}\right) u=λ_{p}\left\vert u(x_{u})\right\vert ^{p-2}u(x_{u})δ_{x_{u}} & \mathrm{in} & Ω\\ u=0 & \mathrm{on} & \partialΩ, \end{array} \right. \] where is the (unique) maximum point of is the Dirac delta distribution supported at \[ \lim_{p\rightarrow\infty}\frac{q(p)}{p}=Q\in\left\{ \begin{array} [c]{lll} (0,1) & \mathrm{if} & N<q(p)<p\\ (1,\infty) & \mathrm{if} & N<p<q(p) \end{array} \right. \] and is such that \[ \min\left\{ \frac{\left\Vert \nabla u\right\Vert _{\infty}}{\left\Vert u\right\Vert _{\infty}}:0\not \equiv u\in W^{1,\infty}(Ω)\cap C_{0}(\overlineΩ)\right\} \leq\lim_{p\rightarrow\infty}(λ_{p})^{\frac{1}{p}}<\infty. \]
27 pages. Some typos were corrected, the final part of the Lemma 4.2's proof was slightly modified