Existence of Three Solutions for Nonlinear Operator Equations and Applications to Second-Order Differential Equations

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.

Liouville-Type Theorem for Nonlinear Elliptic Equations Involving Generalized Greiner Operator

“…Furthermore, the Liouville-type theorem for integral equation and system was established in paper [4]. Other results can be found in [5][6][7][8][9].…”