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| We consider the semilinear Lane-Emden equation on a smooth bounded domain of $\mathbbR^2$ when the exponent $p$ is large. In particular we analyze the asymptotic behavior of sign changing solutions as $p\to+\infty$. Among other results we show, under some symmetry assumptions on the domain, that the positive and negative parts of a family of symmetric solutions concentrate at the same point, as $p\to+\infty$, and the limit profile looks like a tower of two bubbles given by a superposition of a regular and a singular solution of the Liouville problem in $\mathbb R^2$. This is a joint work with I. Ianni and F. Pacella. |
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