| Abstract: |
| We consider the following singularly perturbed elliptic problem
\[
- {\varepsilon ^2}\Delta u + u = f(u){\text{ in }}\Omega ,{\text{ }}u > 0{\text{ in }}\Omega ,{\text{ }}u = 0{\text{ on }}\partial \Omega ,
\]
where $\Omega$ is a domain in ${\mathbb{R}^N}(N \ge 3)$, not necessarily bounded, with boundary $\partial \Omega \in {C^2}$ and the nonlinearity $f$ is of critical growth. In this paper, we construct a family of bound state solutions to the equation given above which concentrates around the local maxima of the distance function from the boundary $\partial \Omega$. |
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