Ground State Solutions for Kirchhoff-type Problems with Critical Nonlinearity

Abstract

In this paper, we study the Kirchhoff-type equation with critical exponent \[ -\left( a + b \int_{\mathbb{R}^3} |\nabla u|^2 \, dx \right) \Delta u + V(x)u = a(x) f(u) + u^5 \quad \textrm{in $\mathbb{R}^3$}, \] where $a,b \gt 0$ are constants, $V \in C(\mathbb{R}^3,\mathbb{R})$, $\lim_{|x| \to \infty} V(x) = V_{\infty} \gt 0$ and $V(x) \leq V_{\infty} + C_1 e^{-b |x|}$ for some $C_1 \gt 0$ and $|x|$ large enough. Via variational methods, we prove the existence of ground state solution.