A uniqueness result for Kirchhoff type problems with singularity

Existence and multiplicity of positive solutions for a class of singular Kirchhoff type problems with sign‐changing potential

“…Recently, Liu et al generalized the results of Lei et al to dimension four. Liao et al obtained the unique result of a class of singular Kirchhoff type problem.…”

“…The Kirchhoff type problem has been extensively studied. For examples, to our best knowledge, Ma and Muñoz Rivera were the first result by the variational method; a quasilinear elliptic equation of Kirchhoff type was considered in Alves et al; the Kirchhoff type problem with critical exponent was first investigated by Alves et al; for the uniqueness result, see Anello and Liao et al; for the multiplicity of solutions for a superlinear Kirchhoff type equations with critical Sobolev exponent in , see Li and Liao; He and Zou considered the infinitely many solutions of the Kirchhoff type problem; the sign‐changing solution was studied by Mao and Zhang, Tang and Cheng and Zhang and Perera; for bound state solutions, see Xie et al; Naimen was the first that considered the Kirchhoff type problem in dimension four; the Kirchhoff type problems was considered by the Yang index in Perera and Zhang. …”

“…(1) When 0 ≤ |g| ∞ ≤ bA 2 according to (5), one has b||u|| 4 −∫ Ω g(x) u 4 |x| s dx ≥ 0, which implies that Φ ′ (t) > 0 for all t > 0. Thus, Φ is strictly increasing for all t > 0.…”

Existence and Uniqueness of Positive Solution for p-Laplacian Kirchhoff-Schrödinger-Type Equation

“…In recent years, a lot of scholars have studied the singular Kirchhoff problem (for more details, we refer the reader to [1][2][3][4]), the Schrödinger-Poisson system (we refer the reader to [5][6][7][8]), and the Kirchhoff-Schrödinger-Poisson system (we refer the reader to [9][10][11][12]). The authors use various methods to obtain the properties of the solution, which makes such problems very interesting.…”

Kirchhoff type equations with strong singularities

“…The singular cases are investigated recently. In 3-dimension, Liu and the first author [37] proved that the nonlinear case f (x)u −γ + λg(x) u p |x| s , with 0 ≤ s < 1, 3 < p < 5 − 2s, −γ > −1 has two positive solutions provided λ > 0 small, and more recently by Liao et al [31] for higher dimension. When −γ < −1 then various difficulties arise in the analysis and it is the purpose of this paper to face them in the Kirchhoff case (1).…”