Solutions with interior bubble and boundary layer for an elliptic problem

Liping Wang; Juncheng Wei

doi:10.3934/dcds.2008.21.333

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2008, Volume 21, Issue 1: 333-351. Doi: 10.3934/dcds.2008.21.333

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Solutions with interior bubble and boundary layer for an elliptic problem

Liping Wang^1, and
Juncheng Wei^2,

1.
Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong
2.
Department of Mathematics, Chinese University of Hong Kong, Shatin, New Territories, Hong Kong

Received: December 31, 2006

Revised: July 31, 2007

Published: December 2007

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Abstract

We study positive solutions of the equation $\varepsilon^2\Delta u - u + u^{\frac{n+2}{n-2}} = 0$, where $n=3,4,5$, and $\varepsilon > 0$ is small, with Neumann boundary condition in a smooth bounded domain $\Omega \subset R^n$. We prove that, along some sequence $\{\varepsilon_j \}$ with $ \varepsilon_j \to 0$, there exists a solution with an interior bubble at an innermost part of the domain and a boundary layer on the boundary $\partial\Omega$.

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Mathematics Subject Classification: Primary: 35B40, 35B45; Secondary: 35J25, 35J67.

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