Normalized solutions to Kirchhoff type equations with a critical growth nonlinearity

Abstract

In this paper, we are concerned with normalized solutions of the Kirchhoff type equation \begin{equation*} -M\left(\int_{\R^N}|\nabla u|^2\mathrm{d} x\right)\Delta u = \lambda u +f(u) \ \ \mathrm{in} \ \ \mathbb{R}^N \end{equation*} with $u \in S_c:=\left\{u \in H^1(\R^N): \int_{\R^N}u^2 \mathrm{d}x=c^2\right\}$. When $N=2$ and $f$ has exponential critical growth at infinity, normalized mountain pass type solutions are obtained via the variational methods. When $N \ge 4$, $M(t)=a+bt$ with $a$, $b>0$ and $f$ has Sobolev critical growth at infinity, we investigate the existence of normalized ground state solutions and normalized mountain pass type solutions. Moreover, the non-existence of normalized solutions is also considered.