Lane Emden problems with large exponents and singular Liouville equations

Abstract

We consider the Lane-Emden Dirichlet problem -Δu = \abs{u}^{p-1}u, in B, u =0, on \partial B, where $p>1$ and $B$ denotes the unit ball in $\IR^2$. We study the asymptotic behavior of the least energy nodal radial solution $u_p$, as $p\to +\infty$. Assuming w.l.o.g. that $u_p(0)<0$, we prove that a suitable rescaling of the negative part $u_p^{-}$ converges to the unique regular solution of the Liouville equation in $\IR^2$, while a suitable rescaling of the positive part $u_p^{+}$ converges to a (singular) solution of a singular Liouville equation in $\IR^2$. We also get exact asymptotic values for the $L^{\infty }$-norms of $u_p^{-}$ and $u_p^{+}$, as well as an asymptotic estimate of the energy. Finally, we have that the nodal line $\N_p:={x\in B : \abs{x}= r_p}$ shrinks to a point and we compute the rate of convergence of $r_p$.

submitted