Asymptotic behavior of solutions to a perturbed \(p\)-Laplacian problem with Neumann condition (Q1779041)

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 \(\partial\Omega\).