Perturbed nonlocal fourth order equations of Kirchhoff type with Navier boundary conditions

Abstract We investigate the existence of multiple solutions for perturbed nonlocal fourth-order equations of Kirchhoff type under Navier boundary conditions. We give some new criteria for guaranteeing that the perturbed fourth-order equations of Kirchhoff type have at least three weak solutions by using a variational method and some critical point theorems due to Ricceri. We extend and improve some recent results. Finally, by presenting two examples, we ensure the applicability of our results

Infinitely many solutions for the stationary Kirchhoff problems involving the fractional p-Laplacian

The aim of this paper is to establish the multiplicity of weak solutions for a Kirchhoff-type problem driven by a fractional p-Laplacian operator with homogeneous Dirichlet boundary conditions

Fractional magnetic Schrödinger-Kirchhoff problems with convolution and critical nonlinearities

In this paper, we are concerned with the existence and multiplicity of solutions for the fractional Choquard-type Schrödinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity: ▫begin{cases} varepsilon^{2s} N([u]^2_{s,A}) (-Delta)^s_A u + V(x)u = (|x|^{-alpha} ast F(|u|^2)) f(|u|^2)u + |u|^{2^ast_s-2}u, & xin mathbb{R}^N, \ U(x) to 0, & text{as} quad |x| to infty, end{cases}▫ where ▫$(-Delta)^s_A$▫ is the fractional magnetic operator with ▫$0 0$▫ is a positive parameter. The electric potential ▫$V in C(mathbb{R}^N, mathbb{R}^+_0)$▫ satisfies ▫$V(x)=0$▫ in some region of ▫$mathbb{R}^N$▫, which means that this is the critical frequency case. We first prove the ▫$(PS)_c$▫ condition, by using the fractional version of the concentration compactness principle. Then, applying also the mountain pass theorem and the genus theory, we obtain the existence and multiplicity of semiclassical states for the above problem. The main feature of our problems is that the Kirchhoff term ▫$M$▫ can vanish at zero