# Parameterized splitting theorems and bifurcations for potential operators, Part I: Abstract theory

@article{Lu2021ParameterizedST, title={Parameterized splitting theorems and bifurcations for potential operators, Part I: Abstract theory}, author={Guangcun Lu}, journal={Discrete \& Continuous Dynamical Systems}, year={2021} }

This is the first part of a series devoting to the generalizations and applications of common theorems in variational bifurcation theory. Using parameterized versions of splitting theorems in Morse theory we generalize some famous bifurcation theorems for potential operators by weakening standard assumptions on the differentiability of the involved functionals, which opens up a way of bifurcation studies for quasi-linear elliptic boundary value problems.

## 3 Citations

### Parameterized splitting theorems and bifurcations for potential operators, Part II: Applications to quasi-linear elliptic equations and systems

- MathematicsDiscrete & Continuous Dynamical Systems
- 2021

<p style='text-indent:20px;'>This is the second part of a series devoting to the generalizations and applications of common theorems in variational bifurcation theory. Using abstract theorems in the…

### A P ] 1 1 N ov 2 02 1 Parameterized splitting theorems and bifurcations for potential operators , Part II : Applications to quasi-linear elliptic equations and systems *

- Mathematics
- 2021

This is the second part of a series devoting to the generalizations and applications of common theorems in variational bifurcation theory. Using abstract theorems in the first part we obtain many new…

### Bifurcations for Hamiltonian systems via dual variational principle

- Mathematics
- 2021

In this paper we study bifurcations for generalized periodic Hamiltonian systems via the ClarkeEkeland dual variational principle and the abstract bifurcation theory developed in author’s previous…

## References

SHOWING 1-10 OF 34 REFERENCES

### Equivariant Bifurcation in Geometric Variational Problems

- Mathematics
- 2014

We prove an extension of a celebrated equivariant bifurcation result of J. Smoller and A. Wasserman [21], in an abstract framework for geometric variational problems. With this purpose, we prove a…

### Morse theory methods for a class of quasi-linear elliptic systems of higher order

- MathematicsCalculus of Variations and Partial Differential Equations
- 2019

We develop the local Morse theory for a class of non-twice continuously differentiable functionals on Hilbert spaces, including a new generalization of the Gromoll-Meyer's splitting theorem and a…

### Homotopy of Extremal Problems: Theory and Applications

- Mathematics
- 2007

This monograph provides a thorough treatment of parameter-dependent extremal problems with local minimum values that remain unchanged under changes of the parameter. The authors consider the theory…

### On the Poincaré-Hopf Theorem for Functionals Defined on Banach Spaces

- Mathematics
- 2009

Abstract Let X be a reflexive Banach space and f : X → ℝ a Gâteaux differentiable function with f′ demicontinuous and locally of class (S)+. We prove that each isolated critical point of f has…

### Critical Point Theory and Hamiltonian Systems

- Mathematics
- 1989

1 The Direct Method of the Calculus of Variations.- 2 The Fenchel Transform and Duality.- 3 Minimization of the Dual Action.- 4 Minimax Theorems for Indefinite Functional.- 5 A Borsuk-Ulam Theorem…

### Bifurcation of Gradient Mappings Possessing the Palais-Smale Condition

- MathematicsInt. J. Math. Math. Sci.
- 2011

This paper considers bifurcation at the principal eigenvalue of a class of gradient operators which possess the Palais-Smale condition and estimates the asyptotic behaviour of solutions to aclass of semilinear elliptic equations with a critical Sobolev exponent.

### The splitting lemmas for nonsmooth functionals on Hilbert spaces II. The case at infinity

- Mathematics
- 2012

We generalize the Bartsch-Li's splitting lemma at infinity for $C^2$-functionals in [2] and some later variants of it to a class of continuously directional differentiable functionals on Hilbert…