Asymptotic behavior of solutions to a perturbed \(p\)-Laplacian problem with Neumann condition
The main goal of this paper is to study the existence and asymptotic behaviour of positive solutions of the following perturbed quasilinear elliptic Neumann problem \[ -\varepsilon\Delta_p u+ |u|^{p-2} u= |u|^{q-2} u\quad\text{in }\Omega,\quad{\partial u\over\partial n}= 0\quad\text{on }\partial\Omega, \] where \(\Delta_p\) denotes the \(p\)-Laplacian operator, \(\Delta_p u= \text{div}(|\nabla u|^{p-2}\,\nabla u)\), \(1< p< q< p^*= {N_p\over N-p}\), \(1< p< N\), \(\varepsilon> 0\) is a parameter, \(\Omega\subset\mathbb{R}^N\) \((N\geq 3)\) is a bounded smooth domain and \(n\) is the outer unit normal to \(\partial\Omega\).
scientific article; zbMATH DE number 2177451
scientific article