Normalized ground-states for the Sobolev critical Kirchhoff equation with at least mass critical growth

Abstract

In this article, we investigate the following nonlinear Kirchhoff equation with Sobolev critical growth: −a+b∫R3∣∇u∣2dxΔu+λu=μf(u)+∣u∣4u,inR3,u>0,∫R3∣u∣2dx=m2,inR3,(Pm)\left\{\begin{array}{l}-\left(a+b\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{3}}{| \nabla u| }^{2}{\rm{d}}x\right)\Delta u+\lambda u=\mu f\left(u)+{| u| }^{4}u,\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3},\\ u\gt 0,\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{3}}{| u| }^{2}{\rm{d}}x={m}^{2},\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3},\end{array}\right.\hspace{2.0em}\hspace{2.0em}\left({P}_{m}) where μ\mu is a positive parameter, a,b>0a,b\gt 0, and the frequency λ\lambda appears as a positive Lagrange multiplier. The nonlinearity ff is more general and satisfies Sobolev subcritical conditions. With the assistance of the Pohožaev constraints and the Sobolev subcritical approximation method, we have achieved a couple of the normalized ground-state solutions to (Pm)\left({P}_{m}) and the asymptotic behavior of the ground-state is also studied.

Keywords

• normalized ground-state solutions
• nonlinear kirchhoff equation
• critical growth
• pohozaev manifold
• 34c37
• 35a15
• 37j45