Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 44, 2019, 1175-1190

EXISTENCE OF POSITIVE SOLUTION FOR THE NONLINEAR KIRCHHOFF TYPE EQUATIONS IN THE HALF SPACE WITH A HOLE

Haiyang He and Xing Yi

Hunan Normal University, College of Mathematics and Statistics
Changsha, Hunan 410081, P.R. China; hehy917 'at' hotmail.com

Hunan Normal University, College of Mathematics and Statistics
Changsha, Hunan 410081, P.R. China; HNyixing522 'at' hotmail.com

Abstract. In this paper, we study the following nonlinear problem of Kirchhoff type

(0.1) –(a + b ∫_Ωr,ρ |∇u|²)Δu + u = |u|^{p – 1}u, u > 0, x ∈ Ω_r,ρ, u ∈ H₀¹(Ω_r,ρ),

where Ω_r,ρ is a half space with a hole which is related to r,ρ in R³, a,b > 0 are constants and 3 < p < 5. We prove that problem (0.1) has a positive high energy solution by using a linking argument with barycenter map restricted on a Nehari manifold.

2010 Mathematics Subject Classification: Primary 58J05, 35J60.

Key words: Kirchhoff type equations, Nehari manifold, positive high energy solution, half space with a hole.

Reference to this article: H. He and X. Yi: Existence of positive solution for the nonlinear Kirchhoff type equations in the half space with a hole. Ann. Acad. Sci. Fenn. Math. 44 (2019), 1175-1190.

Full document as PDF file

https://doi.org/10.5186/aasfm.2019.4462