Normalized Solutions to Kirchhoff Equation with Nonnegative Potential

Abstract

This paper is concerned with the existence of solutions to the problem $$-\left(a+b{\int }_{{\mathbb{R}}^N}|\nabla u{|}^{2}dx\right)\Delta u+V(x)u+\lambda u=|u{|}^{p-2}u,x\in {\mathbb{R}}^N,\lambda \in {\mathbb{R}}^{+}$$ where $a,b>0$ are constants, $V\ge 0$ is a potential, $N\ge 1$, and $p\in (2+\frac{4}{N},2^{\ast }$). We use a more subtle analysis to revisit the limited problem($V\equiv 0$), and obtain a new energy inequality and bifurcation results. Based on these observations, we establish the existence of bound state normalized solutions under different assumptions on $V$. These conclusions extend some known results in previous papers.