Kirchhoff type equations with strong singularities

Abstract

An optimal condition is given for the existence of positive solutions of nonlinear Kirchhoff PDE with strong singularities. A byproduct is that −2 is no longer the critical position for the existence of positive solutions of PDE's with singular potentials and negative powers of the form: −|x| α ∆u = u −γ in Ω, u = 0 on ∂Ω, where Ω is a bounded domain of R N containing 0, with N ≥ 3, α ∈ (0, N ) and −γ ∈ (−3, −1).

Three solutions for a new Kirchhoff-type problem

“…Eq. (1.2) has been studied by many researchers on whole space R N and bounded domain with some boundary conditions, such as [2][3][4][5][6][7][8][9][10][11][12] and their references. Problem (1.2) contains a nonlocal coefficient (a + b Ω |∇u| 2 dx), this leads to that Eq.…”

Existence and uniqueness of solution to a fourth-order Kirchhoff type elliptic equation with strong singularity

“…When α = 0, λ = 1, problem (1.2) becomes a second-order Kirchhoff type equation; these have been studied extensively and many classical results have been obtained in the past few years. For example, Sun et al [ST19] considered problem (1.2) with M (s) = a + bs and f (x, u) = f (x) u γ + g(x)u q , where 0 < f (x) ∈ L 1 (Ω), 0 ≤ g(x) ∈ L ∞ (Ω), q ∈ (0, 1) and γ > 1. By using Ekeland's variational principle on some subset of H 1 0 (Ω) to overcome the difficulty caused by the strongly singular term, they obtained an optimal condition for the existence of positive solutions.…”

Kirchhoff equations with strong singularity in closed manifolds

“…and  denotes the Euclidean Laplace operator, the Kirchhoff-type problem has been extensively investigated. We refer to [3][4][5][6][7][8][9][10][11][12][13][14][15][16] and the references therein for the study of Kirchhoff equations with different kinds of nonlinearities on Euclidean space. The above list is far from being exhaustive.…”

“…in the non-degenerate situation by applying the approximation method. When 1   (strong singularity), an optimal condition in [11] was given for the existence of positive solutions of nonlinear Kirchhoff PDE. In the same direction, we refer to [15] for a uniqueness result involving fractional Laplacian and strong singular nonlinearity.…”

“…Inspired by Wang et al [15] and Sun and Tan [11], the aim of this work is to study the existence and uniqueness of positive solutions to problem (1) with 1   . It is worth noticing that, to the best of our knowledge, the Kirchhoff-type problems involving strong singular nonlinearities on compact Riemannian manifolds have not been studied yet.…”