Existence of solutions for Kirchhoff type equations with unbounded potential

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.

Keywords

• Cut-off functional
• Kirchhoff type problem
• unbounded potential
• variational method