Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent

Multiple positive solutions for Kirchhoff type of problems with singularity and critical exponents

“…There are some recent papers on the Kirchhoff type of problem involving critical exponent, see [21,1,7,11,8]. In particular, in [1], Alves, Corrêa, and Figueiredo have studied the existence of solutions for equations…”

“…After this, Hamydy, Massar and Tsouli [8] extended to the p-Kirchhoff problem. In [21], Xie, Wu and Tang considered the following Kirchhoff type of problem with critical exponent…”

“…Some interesting studies of (1.2) by variational methods can be found. For instance, the existence of positive solutions has been studied extensively in recent years; we refer the reader to [2,14,6,5,18,22,21,13,10] and the references therein. Zhang and Perera [23], Mao and Luan [15], Mao and Zhang [16] have considered sign-changing solutions.…”

Multiplicity of solutions for a fractional Kirchhoff type problem

“…This means that J (u k ), v X0 = 0 for all v ∈ X 0 , that is, J (u k ) = 0 in X * 0 . By (33) and β k (λ n ) → +∞ as k → ∞, we then get that {u k } ∞ k=1 is an unbounded sequence of critical points of J(u). This completes the proof of Theorem 1.3.…”

“…Perera-Zhang [20] studied the existence of nontrivial solutions of problem (5) via the Yang index theory; Zhang-Perera [35] and Mao-Zhang [19] established the existence of sign-changing solutions of problem (5) via invariant sets of descent flow. We refer to [5,6,7,8,15,17,31,33,34] for more existence results of the Kirchhoff type equations.…”