Singularly perturbed nonlinear Dirichlet problems with a general nonlinearity

Byeon, Jaeyoung

doi:10.1090/S0002-9947-09-04746-1

Singularly perturbed nonlinear Dirichlet problems with a general nonlinearity
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by Jaeyoung Byeon

Trans. Amer. Math. Soc. 362 (2010), 1981-2001

DOI: https://doi.org/10.1090/S0002-9947-09-04746-1

Published electronically: November 16, 2009

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Abstract:

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

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