Existence and multiplicity of periodic solutions and subharmonic solutions for a class of elliptic equations

Abstract

This paper focuses on the following elliptic equationwhere the primitive function of f(x, u) is either superquadratic or asymptotically quadratic as |u| → ∞, or subquadratic as |u| → 0. By using variational method, e.g. the local linking theorem, fountain theorem, and the generalized mountain pass theorem, we establish the existence and multiplicity results for the periodic solution and subharmonic solution.

Subharmonic Solutions In Reversible Non-Autonomous Differential Equations

“…The condition (B5) implies that the system (28) is Γ-symmetric. The bifurcation problem (28) with the boundary conditions (29) can be expressed as the following equation…”

“…Then for every αj o ,µo ∈ Λ such that m(µo) is odd we have ω(αj o,µo ) = 0, i.e. the point (αj o ,µo , 0) is a bifurcation point of non-trivial 2πm-periodic solutions for (28).…”

“…Regarding the degree theory (cf. [13,16]), it has been successfully applied to non-Hamiltonian systems in [4,9]) (see also [5,14,28,30]).…”

“…Since the computations of Brouwer G-equivariant degree can be technically challenging, in order to overcome these difficulties we use the equivariant degree package EquiDeg for GAP programming, which was created by Hao-Pin Wu and is available from https://github.com/psistwu/GAP-equideg As the assumption (A3) implies that f (t, 0) = 0, so (1) admits the (trivial) solution u(t) = 0. It is interesting to study a parametrized by α modification of the system (1) (see system (28)) for which the existence of non-constant branches of subharmonic 2πm-periodic solutions bifurcating from 0 can be analyzed. We apply the Brouwer G-equivariant degree method to study the symmetric bifurcation problem for (28).…”

“…It is interesting to study a parametrized by α modification of the system (1) (see system (28)) for which the existence of non-constant branches of subharmonic 2πm-periodic solutions bifurcating from 0 can be analyzed. We apply the Brouwer G-equivariant degree method to study the symmetric bifurcation problem for (28). We establish the existence of multiple branches of subharmonic solutions emerging from the trivial solutions as the parameter α crosses a critical value.…”