Lane Emden problems with large exponents and singular Liouville equations

Abstract

We consider the Lane–Emden Dirichlet problem{−Δu=|u|p−1u,in B,u=0,on ∂B, where p>1 and B denotes the unit ball in R2. We study the asymptotic behavior of the least energy nodal radial solution up, as p→+∞. Assuming w.l.o.g. that up(0)<0, we prove that a suitable rescaling of the negative part up− converges to the unique regular solution of the Liouville equation in R2, while a suitable rescaling of the positive part up+ converges to a (singular) solution of a singular Liouville equation in R2. We also get exact asymptotic values for the L∞-norms of up− and up+, as well as an asymptotic estimate of the energy. Finally, we have that the nodal line Np:={x∈B:|x|=rp} shrinks to a point and we compute the rate of convergence of rp.