On an elliptic Kirchhoff-type problem depending on two parameters

In this paper, we consider the Dirichlet problem associated to an elliptic Kirchhoff-type equation depending on two parameters. Under rather general and natural assumptions, we prove that, for certain values of the parameters, the problem has at least three solutions

Existence and Multiplicity of solutions for Kirchhoff type equations in Physical Education

The fact that potentially skilled, but biologically later-maturing athletes are less likely to be selected into talent development programmes can represent a failure of Talent Identification in sports. In this article, we prove the existence of solutions for Kirchhoff type equations with Dirichlet boundaryvalue condition. We use the Mountain Pass Theorem in critical point theory, without the (PS) condition

Existence and multiplicity results for a coupled system of Kirchhoff type equations

This paper deals with a coupled system of Kirchhoff type equations in ${\mathbb{R}}^{3}$. Under suitable assumptions on the potential functions $V(x)$ and $W(x)$, we obtain the existence and multiplicity of nontrivial solutions when the parameter $\lambda$ is sufficiently large. The method combines the Nehari manifold and the mountain-pass theorem

Ground state solution of a nonlocal boundary-value problem

In this paper, we apply the method of the Nehari manifold to study the Kirchhoff type equation \begin{equation*} -\Big(a+b\int_\Omega|\nabla u|^2dx\Big)\Delta u=f(x,u) \end{equation*} submitted to Dirichlet boundary conditions. Under a general $4-$superlinear condition on the nonlinearity $f$, we prove the existence of a ground state solution; that is a nontrivial solution which has least energy among the set of nontrivial solutions. In case which $f$ is odd with respect to the second variable, we also obtain the existence of infinitely many solutions. Under our assumptions the Nehari manifold does not need to be of class $\mathcal{C}^1$.Comment: 8 page