Spike-layered solutions with compact support to some singularly perturbed quasilinear elliptic problems in general smooth domains. (Q1421235)

The authors deal with the following singularly perturbed problem \[ \begin{cases} -\varepsilon\Delta_pu= f(u)\quad &\text{in }\Omega,\\ u\geq 0\text{ in }\Omega,\;u= 0\quad &\text{on }\partial\Omega\end{cases}\tag{1} \] and study the structure of solutions for (1) as \(\varepsilon\to 0\), where \(\Delta_p u= \text{div}(D| Du|^{p-2}Du)\) with \(p> 1\), and \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^N\) \((N\geq 1)\). Using of the sub- and supersolution method, the author proves under suitable assumptions on \(f\) existence of many solutions for (1) an show that they are spike-layered solutions. Moreover, the measure of each spike layer is estimated as \(\varepsilon\to 0\).