On the existence of positive solution for an elliptic equation of Kirchhoff type via Moser iteration method

Abstract

Dedicated to our dear friend and collaborator Professor Claudianor O. AlvesWe investigate the questions of existence of positive solution for the nonlocal problem −M( u 2 )Δu = f (λ,u) in Ω and u = 0 on ∂Ω, where Ω is a bounded smooth domain of R N , and M and f are continuous functions.

A sub-supersolution approach for some classes of nonlocal problems involving Orlicz spaces

“…For more inquiries on nonlocal problems we refer to [13,6] where systems of elliptic equations are examined. With regard to the scalar case, we quote the papers [3,5,11,25,27,28,36]. Such problems are important for applications in view of the significant number of physical phenomena formulated into nonlocal mathematical models.…”

On an elliptic equation ofp-Kirchhoff type via variational methods

“…Motivated by papers [2,5] and by some ideas developed in [3,6], we prove the existence of positive solutions to (P) and (P)A-However, in this work, we use a different approach to those explored in [2,5,3], because here we are working with the p-Laplacian operator. Some estimates for this type of operator can not be obtained using the same kind of ideas explored for the case p=2.…”

“…Here, we use the version due to Willem for the Mountain Pass Theorem (see [15, p. 12]). [5] Existence of solutions 267 LEMMA 2 . 2 .…”

On fractional regularity methods for a class of nonlocal problems

“…Lions in [32], where the original hyperbolic version of (P) is investigated. Although that it is remarkable that Kirchhoff problems are related to the modelling of the nonlinear vibration phenomenon, for the sake of brevity, we refer the reader to [1,9,10,24,32,35,42] and the references thereof for further information on this subject, where the physical background is discussed in more detail.…”