Ground State Solutions for Kirchhoff Equations via Modified Nehari-Pankov Manifold

Abstract

We investigate the Kirchhoff type elliptic problem

$\left(a+b \int_{\mathbb{R}^{N}}\left[|\nabla u|^{2}+V(x) u^{2}\right] \mathrm{d} x\right)[-\Delta u+V(x) u]=f(x, u), \quad x \in \mathbb{R}^{N}$

where both V and f are periodic in x, 0 belongs to a spectral gap of −Δ+V. Under suitable assumptions on V and f with more general conditions, we prove the existence of ground state solutions and infinitely many geometrically distinct solutions.

Keywords

Kirchhoff equation / Nehari-Pankov manifold / ground state solution / multiplicity of solutions