Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 44, 2019, 987-1002

EXISTENCE OF POSITIVE SOLUTION FOR KIRCHHOFF TYPE PROBLEM WITH CRITICAL DISCONTINUOUS NONLINEARITY

Giovany M. Figueiredo and Gelson G. dos Santos

Universidade de Brasilia - UNB, Departamento de Matemática
CEP. 70910-900 , Brasília - DF, Brazil; giovany 'at' unb.br

Universidade Federal do Pará - UFPA, Faculdade de Matemática
CEP. 6075-110, Belém - PA, Brazil; gelson 'at' ymail.com

Abstract. In this paper we are concerned with existence of positive solution to the class of nonlinear problems of the Kirchhoff type given by

L_ε(u) = H(u – β)f(u) + u^2*-1 in R^N,
u ∈ H¹(R^N ∩ W^2,q/(q-1)(R^N),

where N ≥ 3, q ∈ (2,2*), ε,β > 0 are positive parameters, f : R → R is a continuous function, H is the Heaviside function, i.e., H(t) = 0 if t ≤ 0, H(t) = 1 if t > 0 and

L_ε(u) := [M(1/ε^N-2 ∫_RN|∇u|² dx + 1/ε^N ∫_RN V(x)|u|² dx)][–ε²Δu + V(x)u].

The function M is a general continuous function. The function V is a positive potential that satisfies following hypothesis: or V satisfies the Palais–Smale condition or there is a bounded domain Ω in R^N such that V has no critical point in ∂Ω. Here we use a suitable truncation to apply a version of the penalization method of Del Pino and Felmer [16] combined with the Mountain Pass Theorem for locally Lipschitz functional.

2010 Mathematics Subject Classification: Primary 35A15, 35B33, 35B25, 35J60.

Key words: Variational methods, critical exponents, Kirchhoff equation, discontinuous nonlinearity.

Reference to this article: G. M. Figueiredo and G. G. dos Santos: Existence of positive solution for Kirchhoff type problem with critical discontinuous nonlinearity. Ann. Acad. Sci. Fenn. Math. 44 (2019), 987-1002.

Full document as PDF file

https://doi.org/10.5186/aasfm.2019.4453