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Solutions with interior bubble and boundary layer for an elliptic problem

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  • 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 $\partial\Omega$.
    Mathematics Subject Classification: Primary: 35B40, 35B45; Secondary: 35J25, 35J67.


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