Narrow Results

In this paper, by Morse theory we will study the Kirchhoff type equation with an additional critical nonlinear term, and the main results are to compute the critical groups including the cases where zero is a mountain pass solution and the nonlinearity is resonant at zero. As an application, the multiplicity of nontrivial solutions for this equation with the parameter across the first eigenvalue is investigated under appropriate assumptions. To our best knowledge, estimates of our critical groups are new even for the Kirchhoff type equations with subcritical nonlinearities.

In this paper, we deal with fractional $p$-Laplace Kirchhoff equations with exponential growth of the form

In this paper, we use the genus properties in critical point theory to prove the existence of infinitely many solutions for fractional $p$-Laplacian equations of Schrödinger-Kirchhoff type.

Everyday experience shows that twisted elastic filaments spontaneously form loops. We model the dynamics of this looping process as a sequence of bifurcations of the solutions to the Kirchhoff equation describing the evolution of thin elastic filaments. The control parameter is taken to be the initial twist density in a straight rod. The first bifurcation occurs when the twisted straight rod deforms into a helix. This helix is an exact solution of the Kirchhoff equations, whose stability can be studied. The secondary bifurcation is reached when the helix itself becomes unstable and the localization of the post-bifurcation modes is demonstrated for these solutions. Finally, the tertiary bifurcation takes place when a loop forms at the middle of the rod and the looping becomes ineluctable. Emphasis is put on the dynamical character of the phenomena by studying the dispersion relation and deriving amplitude equations for the different configurations.