Periodic and homoclinic solutions of some semilinear sixth-order differential equations
The solvability of the boundary value problem \((P)\) \[ u^{vi}+Au^{iv}+Bu''+u-u^3=0, \quad 0<x<L, \] \[ u(0)=u''(0)=u^{iv}(0)=0, \quad u(L)=u''(L)=u^{iv}(L)=0, \] is considered, where \(A\) and \(B\) are positive constants such that \(A^{2}<4B\). The existence of a nontrivial solution follows by applying a minimization theorem and a multiplicity result based on Clark's theorem. The mountain-pass theorem of Brezis-Nirenberg and concentration-compactness arguments are used to establish the existence of homoclinic solutions to \[ u^{vi}+A^{iv}+Bu''-u+a(x)u|u|^{\sigma}=0, \] where \(a\) is positive periodic function and \(\sigma\) is a positive constant.
scientific article; zbMATH DE number 1836786