Abstract
Abstract We give a new non-smooth Clark’s theorem without the global symmetric condition. The theorem can be applied to generalized quasi-linear elliptic equations with small continous perturbations. Our results improve the abstract results about a semi-linear elliptic equation in Kajikiya [10] and Li-Liu [11].
Highlights
The Clark’s theorem is a important result in critical point theory
The theorem can be applied to generalized quasi-linear elliptic equations with small continous perturbations
Our results improve the abstract results about a semi-linear elliptic equation in Kajikiya [10] and Li-Liu [11]
Summary
The Clark’s theorem is a important result in critical point theory (see [4, 8]). Using this theorem for the even coercive functional, the existence of a sequence of negative critical values tending to is obtained. In [3] Chen-Liu-Wang showed a version of the Clark’s theorem without the Palais-Smale conditions ((P-S) conditions) They studied the existence of in nitely many solutions for a degenerate quasi-linear elliptic operator and a second-order Hamiltonian system via their abstract theory. All those versions of Clark’s theorem references to above rely on the symmetric condition about the Euler-Lagrange functional. In [10], Kajikiya established the existence of in nitely many critical points about C functional without the global symmetric condition As applications, they obtained the existence of in nitely many solutions of the sub-linear elliptic equation with a small perturbation.
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