Lane Emden problems with large exponents and singular Liouville equations

Abstract

We consider the Lane-Emden Dirichlet problem −∆u = |u| p−1 u, in B,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 → +∞. 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 ∞ -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 ∈ B : |x| = r p } shrinks to a point and we compute the rate of convergence of r p .

The Hénon problem with large exponent in the disc

“…This Theorem extends already known results for the Lane Emden equation to the Hénon problem. Surprisingly the limit values of the maximum and the minimum of the radial solutions u i p are not affected at all by the term |x| α and they are exactly the same of the Lane Emden case which have been characterized in [AG] and [GGP2]. The dependence on the parameter α appears instead in the limit of the two rescaling U α,δ(α) and Z α,γ:δ2(α) .…”

“…In the following we shall write Z ℓ = Z γ(ℓ);δ(ℓ) for such function. Notice that the parameter H in the notation used in [GGP2] is identified by the relation…”

“…In this paper, carrying on the study of the Hénon problem started in [AG2,AG3], we consider (1.1) in the unit disc, where it admits solutions for every p > 1 and no critical exponent appears and in particular we focus on large values of p, where concentration phenomena take place and nonradial positive solutions arise. Indeed it has been shown in [AG] and [GGP2], both dealing with the Lane Emden problem, that radial solutions behave like a spike. Such kind of concentration differs from the one occurring in the higher dimensional case when p approaches the critical exponent: firstly because solutions stay bounded, secondly because the concentration of nodal solutions follows different paths (and has different limit problems), depending on the nodal zone which is focused.…”

Sign-changing tower of bubbles for a sinh-Poisson equation with asymmetric exponents

“…Since the appropriate choice of α i 's leads to α i = α j for i = j, we find that the bubble tower approximate solution W ρ is actually the sum of solutions to different singular Liouville problems. Such new blow-up profiles were observed in [6]. Towers of concentrated solutions to different singular Liouville equations were initially introduced in the article [7], where the case γ = 1 is considered, and which is the main motivation to this work.…”

Quasi-radial solutions for the Lane–Emden problem in the ball

“…Let us first recall that u p admits two limit problems, one obtained when rescaling u p with respect to its maximum point, which is 0 (the scaling parameter in this case is ε + p defined by ε + p −2 = pu p (0) p−1 ) and the second one obtained rescaling u p with respect to its minimum point s p (with scaling parameter given by ε − p ) (see [GGP,Theorem 1] for the rigorous statement of the result).…”