Lane Emden problems with large exponents and singular Liouville\n equations

2012/09/07 by Massimo Grossi, Grossi, Massimo, Christopher Grumiau +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1209.1534

openalex publication_date 2012/09/07 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

We consider the Lane-Emden Dirichlet problem -\Δ u = absu^p-1u, in\nB, u =0, on \∂ B, where p>1 and B denotes the unit ball in IR².\nWe study the asymptotic behavior of the least energy nodal radial solution\nu_p, as p\→ +\∞. Assuming w.l.o.g. that u_p(0) < 0, we prove\nthat a suitable rescaling of the negative part u_p^- converges to the unique\nregular solution of the Liouville equation in IR², while a suitable\nrescaling of the positive part u_p⁺ converges to a (singular) solution of a\nsingular Liouville equation in IR². We also get exact asymptotic values for\nthe L^\∞-norms of u_p^- and u_p⁺, as well as an asymptotic estimate\nof the energy. Finally, we have that the nodal line N_p:=x\∈ B : absx=\nr_p shrinks to a point and we compute the rate of convergence of r_p.\n

Citations

• Existence of solitary waves in higher dimensions

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