Concentration of solutions for some singularly perturbed mixed problems: existence results
Let \(\Omega\) be a bounded open subset of \(\mathbb R^n\) with smooth boundary. Let \(p\in \,]1,\frac{n+2}{n-2}[\). Let \(\nu\) be the outward unit normal to \(\partial\Omega\). Let \(\partial_{\mathcal D}\Omega\), \(\partial_{\mathcal N}\Omega\) be disjoint open subsets of the boundary \(\partial\Omega\) of \(\Omega\) such that the union of the closures of \(\partial_{\mathcal D}\Omega\) and of \(\partial_{\mathcal N}\Omega\) equals \(\partial\Omega\). The authors show that under suitable assumptions on the geometry of \(\partial\Omega\), if \(\overline{Q}\) belongs to the intersection of the closures of \(\partial_{\mathcal D}\Omega\) and of \(\partial_{\mathcal N}\Omega\), then the mixed boundary value problem \[ -\varepsilon^2\Delta u + u=u^p\quad\text{in }\Omega, \qquad u>0 \quad\text{on }\Omega, \qquad \frac{\partial u}{\partial \nu}=0\quad\text{on }\partial_{\mathcal N}\Omega, \qquad u=0\quad\text{on }\partial_{\mathcal D}\Omega, \] admits a family of solutions \(u_\varepsilon\) which concentrate at \(\overline{Q}\) as the parameter \(\varepsilon>0\) shrinks to \(0\).
singularly perturbed nonlinear Poisson equation
scientific article; zbMATH DE number 5731030
scientific article