Lane Emden problems with large exponents and singular Liouville equations

We consider the Lane-Emden Dirichlet problem [GRAPHICS] where p > 1 and B denotes the unit ball in R-2. We study the asymptotic behavior of the least energy nodal radial solution u(p), as p -> +infinity. 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 R-2, while a suitable rescaling of the positive part u(p)(+) converges to a (singular) solution of a singular Liouville equation in R-2. We also get exact asymptotic values for the L-infinity-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 is an element of B: vertical bar x vertical bar = r(p)} shrinks to a point and we compute the rate of convergence of r(p). (C) 2014 Published by Elsevier Masson SAS.