Electron. J. Differential Equations, Vol. 2017 (2017), No. 184, pp. 1-12.

Existence of solutions for Kirchhoff type equations with unbounded potential

Yueliang Duan, Yinggao Zhou

Abstract:
In this article, we study the Kirchhoff type equation
$$ \Big(a+\lambda\int_{\mathbb{R}^3}|\nabla u|^2 +\lambda b\int_{\mathbb{R}^3}u^2\Big)[-\Delta u+b u] =K(x)|u|^{p-1}u,\quad \text{in } \mathbb{R}^3, $$
where $a,b>0$, $p\in(2,5)$, $\lambda\geq0$ is a parameter, and K may be an unbounded potential function. By using variational methods, we prove the existence of nontrivial solutions for the above equation. A cut-off functional and some estimates are used to obtain the bounded Palais-Smale sequences.

Submitted March 7, 2017. Published July 19, 2017.
Math Subject Classifications: 35A15, 35A20, 35J20.
Key Words: Cut-off functional; Kirchhoff type problem; unbounded potential; variational method.

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Yueliang Duan
School of Mathematics and Statistics
Central South University
Changsha, Hunan 410083, China
email: duanyl@csu.edu.cn
Yinggao Zhou
School of Mathematics and Statistics
Central South University
Changsha, Hunan 410083, China
email: ygzhou@mail.csu.edu.cn

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