| Abstract: |
| In this talk, we are concerned with the Lane-Emden system $-Delta u = v^p$ and $-Delta v = u^q$posed on bounded domain with the Dirichlet zero boundary condition. For this system, if $p$ and $q$ satisfies $\frac{1}{p+1} + \frac{1}{q+1} = \frac{n}{n-2} -\ep$ with very small $ep>0$, then it is believed that most of the solutions are usually spike-shaped and their asymptotic behavior can be described as it was done for the single equation $-\Delta u = u^{(n+2)/(n-2)-\ep}$. However, due to the technical difficulty, this type of analysis has been verified only for the energy-minimizing solutions by Guerra (08`) with an additional assumption that $\min{p,q} \geq 1$ and $\Omega$ is convex, and he conjectured that the same asymptotic behavior would be true without the assumption $\min{p,q} \geq 1$. In our work, we settle this conjecture and also remove the convexity condition on the domain. This is based on a joint work with Seunghyeok Kim (KIAS). |
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