Abstract
In this paper, we study the existence of normalized ground state solution to the following Kirchhoff-type equation
\(-\left(a+b\int\limits_{\mathbb{R}^{N}}|\nabla u|^{2}\text{d}x\right)\Delta u+(V(x)+\mu)u=|u|^{p-2}u\quad\text{in }\mathbb{R}^{N}\)
under the constraint \(\int_{\mathbb{R}^{N}}|u|^{2}\text{d}x=c^{2}\) , where \(a,b,c>0\) , \(p\in\left(2,\frac{2N+8}{N}\right)\) with \(N=1,2,3\) , \(\mu\in\mathbb{R}\) is unknown and appears as a Lagrange multiplier, and \(V:\mathbb{R}^{N}\to[0,+\infty)\) is bounded and continuous. We apply the Gagliardo–Nirenberg inequality to obtain the boundedness from below of the energy functional and the boundedness of the minimizing sequence. The existence of ground state normalized solution to the above equation is established associated with different cases of potential \(V(x)\) . Moreover, the corresponding results for the fractional Kirchhoff-type equation are also true by using the fractional version of Gagliardo–Nirenberg inequality.
