Electron. J. Diff. Equ., Vol. 2016 (2016), No. 102, pp. 1-22.

Nontrivial solutions for Kirchhoff equations with periodic potentials

Xiaoyan Ma, Xiaoming He

Abstract:
In this article we consider the Kirchhoff equations
$$ -\Big(a+b\int_{\mathbb{R}^3}|\nabla u|^2\Big)\Delta u+V(x)u=f(x,u),\quad x\in\mathbb{R}^3, $$
where $a,b>0$ are constants, the nonlinearity f is superlinear at infinity with subcritical or critical growth and V is positive, continuous and periodic in x. Some existence results for ground state solutions are obtained by using variational methods. Moreover, when $V\equiv 1$ we obtain ground state solutions for the above problem with a wide class of superlinear nonlinearities by using a new approach.

Submitted March 28, 2016. Published April 20, 2016.
Math Subject Classifications: 47G20, 35J50, 35B65.
Key Words: Kirchhoff-type problems; ground states; Nehari manifold; critical Sobolev exponent.

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Xiaoyan Ma
College of Science
Minzu University of China
Beijing 100081, China
email: SX140963@muc.edu.cn
Xiaoming He
College of Science
Minzu University of China
Beijing 100081, China
email: xmhe923@muc.edu.cn

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