Singularly perturbed nonlinear Dirichlet problems with a general nonlinearity

Abstract

Let Ω \Omega be a bounded domain in R n , \mathbf {R}^n, n ≥ 3 , n \ge 3, with a boundary ∂ Ω ∈ C 2 . \partial \Omega \in C^2. We consider the following singularly perturbed nonlinear elliptic problem on Ω \Omega : \[ ε 2 Δ u − u + f ( u ) = 0 ,     u > 0  on  Ω , u = 0  on  ∂ Ω , \varepsilon ^2 \Delta u - u + f(u) = 0, \ \ u > 0 \textrm { on }\Omega , \quad u = 0 \textrm { on } \partial \Omega , \] where the nonlinearity f f is of subcritical growth. Under rather strong conditions on f , f, it has been known that for small ε > 0 , \varepsilon > 0, there exists a mountain pass solution u ε u_\varepsilon of above problem which exhibits a spike layer near a maximum point of the distance function d d from ∂ Ω \partial \Omega as ε → 0. \varepsilon \to 0. In this paper, we construct a solution u ε u_\varepsilon of above problem which exhibits a spike layer near a maximum point of the distance function under certain conditions on f f , which we believe to be almost optimal.