Behavior near criticality of sign-changing solutions to the Lane-Emden equation with Dirichlet and Neumann boundary conditions

Abstract: In this talk we will be concerned with several recent results on the classical Lane-Emden equation in a bounded domain, with either Dirichlet or Neumann boundary conditions. In the first part of the talk, for radial solutions and in the case of the ball in dimension 3 or higher, we provide sharp rates and constants describing the asymptotic behavior (as we approach the Sobolev critical exponent) of all local minima and maxima of the solutions, as well as its derivative at roots. As corollaries, we complement a known asymptotic approximation of the Dirichlet nodal solution in terms of a tower of bubbles and present a similar formula for the Neumann problem. In the second part or the talk we analyse the nonradial case with Neumann boundary conditions, namely the existence (and symmetry) of least energy solutions and their dependence on the exponent of the nonlinearity up to the Sobolev critical exponent, discussing also the slightly supercritical case. The talk is based on joint works with Alberto Saldaña, Angela Pistoia and Massimo Grossi.

analysis of PDEs

Audience: researchers in the topic

Comments: To join Hugo's talk, please use the following information:

Link to join Hugo's talk: puc-rio.zoom.us/j/96415145570?pwd=TERzRmp1Z2RyU2x6OU92dFdHUnBmZz09

Meeting ID: 964 1514 5570 Password: 125966

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Rio de Janeiro webinar on analysis and partial differential equations

Series comments: Talks are held twice a month; start time and day of the week may vary, according to the speaker time zone. A link to join each webinar will be made available in due time here and at sites.google.com/view/webinarpde/home

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