Abstract
The existence of three solutions for nonlinear operator equations is established via index theory for linear self-adjoint operator equations, critical point reduction method, and three critical points theorems obtained by Brezis-Nirenberg, Ricceri, and Averna-Bonanno. Applying the results to second-order Hamiltonian systems satisfying generalized periodic boundary conditions or Sturm-Liouville boundary conditions and elliptic partial differential equations satisfying Dirichlet boundary value conditions, we obtain some new theorems concerning the existence of three solutions.
Highlights
Introduction and Main ResultsIn their excellent paper [1] in 1991, Brezis-Nirenberg proved the existence of two nonzero critical points under suitable assumptions by a negative gradient flow and linking
The obtained result was applied to second-order Hamiltonian systems and yielded the existence of three periodic solutions for these systems
In [3], one new existence of at least three periodic solutions for second-order Hamiltonian systems was obtained by using the critical point reduction method and two nonzero critical points theorem in [1, 4]
Summary
In their excellent paper [1] in 1991, Brezis-Nirenberg proved the existence of two nonzero critical points under suitable assumptions by a negative gradient flow and linking. In [2], Tang obtained one existence of at least three periodic solutions for second-order Hamiltonian systems by using two nonzero critical points theorem in [1]. This result generalizes the corresponding result in [1]. In [3], one new existence of at least three periodic solutions for second-order Hamiltonian systems was obtained by using the critical point reduction method and two nonzero critical points theorem in [1, 4]. We give some examples and some remarks to illustrate that the corresponding results of [1,2,3, 9, 10, 15, 22] are special cases of these results in a sense
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