Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 53, pp. 1-12.

Title: Kirchhoff-type problems with critical Sobolev exponent in a hyperbolic space
       
Authors: Paulo Cesar Carriao (Univ. Federal de Minas Gerais, Belo Horizonte, MG, Brazil)
         Augusto Cesar dos Reis Costa (Univ. Federal do Para, Belem, PA, Brazil)
         Olimpio Hiroshi Miyagaki (Univ. Federal de Juiz de Fora, MG, Brazil)
         Andre Vicente (Univ. Estadual do Oeste do Parana, Cascavel, PR, Brazil)

Abstract: 
 In this work we study a class of the critical Kirchhoff-type problems in a Hyperbolic space.
 Because of the Kirchhoff term, the nonlinearity u<sup>q</sup> becomes concave for 2&lt;q&lt;4,
 This brings difficulties when proving the boundedness of Palais Smale sequences.
 We overcome this difficulty by using a scaled functional related with a Pohozaev manifold.
 In addition, we need to overcome  singularities on the unit sphere, so that we use
 variational methods to obtain our results.

Submitted January 30, 2021. Published June 14, 2021.
Math Subject Classifications: 58J05, 35R01, 35J60, 35B33.
Key Words: Kirchhoff-type problem; variational methods; hyperbolic space.