Singularly perturbed nonlinear Dirichlet problems with a general nonlinearity

  • Byeon J
30Citations
Citations of this article
12Readers
Mendeley users who have this article in their library.

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.

Cite

CITATION STYLE

APA

Byeon, J. (2009). Singularly perturbed nonlinear Dirichlet problems with a general nonlinearity. Transactions of the American Mathematical Society, 362(4), 1981–2001. https://doi.org/10.1090/s0002-9947-09-04746-1

Register to see more suggestions

Mendeley helps you to discover research relevant for your work.

Already have an account?

Save time finding and organizing research with Mendeley

Sign up for free