Electron. J. Diff. Equ., Vol. 2015 (2015), No. 125, pp. 1-10.

Existence and multiplicity of solutions for sublinear ordinary differential equations at resonance

Chengyue Li, Fenfen Chen

Abstract:
Using a $Z_2$ type index theorem, we show the existence and multiplicity of solutions for the sublinear ordinary differential equation
$$ \mathcal{L} u(t)=\mu u(t)+W_u(t,u(t)),\quad 0\leq t\leq L $$
with suitable periodic or boundary conditions. Here $\mathcal{L}$ is a linear positive selfadjoint operator, $\mu$ is a parameter between two egienvalues of this operator, and $W_u$ is the gradient of a potential function.

Submitted February 5, 2015. Published May 6, 2015.
Math Subject Classifications: 58E05, 34C37, 70H05.
Key Words: Sublinear potential; $Z_2$ type index theorem; critical point; resonance; Hamiltonian system.

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Chengyue Li
Department of Mathematics
Minzu University of China
Beijing 100081, China
email: cunlcy@163.com
Fenfen Chen
Department of Mathematics
Minzu University of China
Beijing 100081, China
email: chenfenfen359@163.com

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