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Discrete and Continuous Dynamical Systems - Series A (DCDS-A)
 

Solutions with interior bubble and boundary layer for an elliptic problem

Pages: 333 - 351, Volume 21, Issue 1, May 2008      doi:10.3934/dcds.2008.21.333

 
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Liping Wang - Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong, China (email)
Juncheng Wei - Department of Mathematics, Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, China (email)

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 $\partial\Omega$.

Keywords:  Semilinear elliptic problem, critical Sobolev exponent, blow up.
Mathematics Subject Classification:  Primary: 35B40, 35B45; Secondary: 35J25, 35J67.

Received: January 2007;      Revised: August 2007;      Available Online: February 2008.