On the shape of the nonnegative solutions to a singularly perturbed quasilinear Dirichlet problem (Q1413956)

The author studies the shape of the nontrivial nonnegative solutions for a singularly perturbed quasilinear Dirichlet problem with non-Lipschitz nonlinearity \[ -\varepsilon \text{ div }(|Du|^{p-2}Du) =f(u)\quad\text{ in }\Omega,\qquad u|_{\partial\Omega}=0. \] Here \(\varepsilon>0\) is small enough, \(p>1\) and \(\Omega\in {\mathbb R}^n,\) \(n\geq 2\), is a bounded domain with smooth boundary. It is shown that there exists at least one nontrivial nonnegative solution of that problem which consists of some bubble. The profile of each bubble is also discussed.